math nation algebra 1 homework

Math Nation

About this app

Data safety.

Ratings and reviews

• Flag inappropriate
• Show review history

What's new

App support.

If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

To log in and use all the features of Khan Academy, please enable JavaScript in your browser.

Get ready for Algebra 1

Unit 1: get ready for equations & inequalities, unit 2: get ready for working with units, unit 3: get ready for linear relationships, unit 4: get ready for functions & sequences, unit 5: get ready for exponents, radicals, & irrational numbers, unit 6: get ready for quadratics.

Free Algebra Worksheets

Algebra Worksheets

All of the Algebra 1 Worksheets below are samples from the Algebra Worksheets Library on our Infinite K-8 Math Worksheet Portal .

Click any of the links below to download the corresponding Algebra Math Worksheets and answer key.

▶️: Sample Worksheet Download | 🔒: Worksheet Only Available to Members | 🔽 Jump to a Topic:

Foundational Algebra Skills

Inequalities

Exponents and Scientific Notation

Linear Functions

Radicals and Radical Formulas

Solving Systems

Working with Polynomials

Rational Expressions

Statistics/Data

Solving Algebraic Equations

Word Problems

Welcome to the official Mashup Math Algebra Worksheets library. Below you will find a complete collection of printable Algebra 1 Worksheets organized by topic. Every worksheet can be downloaded as a PDF file that easy to print and/or share on online learning platforms such as Google Classroom,

Our topic-based Algebra Worksheets were carefully designed with the needs of Algebra students in mind. Our Algebra 1 Worksheets library covers a complete set of Algebra (and Pre-Algebra) topics that would commonly be featured in a curriculum for an Algebra course or online program. This library includes worksheets that will allow you to practice common Algebra topics such as working with exponents, solving equations, inequalities, solving and graphing functions, systems of equations, factoring, quadratic equations, algebra word problems, and more.

All of our algebra worksheets were created by math educators with the aim of helping students to practice and learn important algebra skills while being appropriately challenged at the same time. This way, algebra students can strengthen their skills and/or identify weaknesses in an appropriate and effective way. If you are an algebra teacher, tutor, or specialist, you will find our Algebra 1 worksheets to be an effective resource that will come in handy time and time again.

Consider a math teacher who was helping her students how to find the slope of a line from a given graph. After teaching a lesson on this topic and demonstrating the procedure, she could then assign her students one of our algebra worksheets on finding slope from a graph. While students practice this new skill, the teacher can move about the classroom and offer small group instruction/support. Rather than spending an entire class period lecturing, the teacher was able to give her students an effective and meaningful opportunity to apply, practice, and assess a new math skill and procedure.

Algebra teachers who use our topic-specific worksheets often learn quickly that Mashup Math learning resources are not only effective, but engaging to students. Unlike many free algebra 1 worksheets available online, our worksheets are designed with an emphasis on providing students with meaningful and effective learning opportunities, which is why we only include an age-appropriate amount of practice problems on each worksheet. This quality over quantity approach prevents students from growing bored and disinterested with learning math in a monotonous and overly repetitive way.

If you are new to Mashup Math, we strongly suggest that you scroll through the Free Algebra Worksheets Library below and download a few worksheets related to whatever topic you are currently working on with your students (complete answer keys are included). Once you share these activities with your students, you will have a better idea of why our resources can be so effective and engaging.

And if you would like to gain access to our full library of Algebra 1 Math Worksheets, then go ahead and click on the link below to sign-up for annual access to the Mashup Math Infinite Math Worksheet Portal , where users gain on-demand access to ALL of our topic-specific math worksheets for all algebra and geometry topics.

🛑 WAIT! Would you like FREE math activities, lesson resources, worksheets, and puzzles dropped in your inbox every week? 💁‍♀️

⇥ Click here to join our mailing list and get a free pdf math workbook as a bonus!

▶️ Commutative property of Multiplication (A)

🔒 Commutative property of Multiplication (B)

▶️ Associative property of Multiplication (A)

🔒 Associative property of Multiplication (B)

▶️ Practice with the Distributive Property (Algebra) (A)

🔒 Practice with the Distributive Property (Algebra) (B)

🔒 Practice with the Distributive Property (Algebra) (C)

▶️ Practice with Absolute Values (A)

🔒 Practice with Absolute Values (B)

▶️ Practice: Order of Operations with Exponents (A)

🔒 Practice: Order of Operations with Exponents (B)

🔒 Practice: Order of Operations with Exponents (C)

▶️ Practice: Order of Operations with Nested Parenthesis (A)

🔒 Practice: Order of Operations with Nested Parenthesis (B)

🔒 Practice: Order of Operations with Nested Parenthesis (C)

▶️ Extra practice (order of operations up to 6 terms) (A)

🔒 Extra practice (order of operations up to 6 terms) (B)

▶️ Extra practice (order of operations up to 6 terms) (C)

▶️ Extended Practice: Order of Operations

▶️ Finding percent increase/decrease (A)

🔒 Finding percent increase/decrease (B)

▶️ Solving simple proportions (A)

🔒 Solving simple proportions (B)

🔒 Solving simple proportions (C)

▶️ Simplifying ratios (A)

🔒 Simplifying ratios (B)

▶️ Similar Triangles (A)

🔒 Similar Triangles (B)

▶️ Factoring numbers (up to 100) (A)

🔒 Factoring numbers (up to 100) (B)

🔒 Factoring numbers (up to 100) (C)

▶️ Finding Greatest Common Factor (GCF) (A)

🔒 Finding Greatest Common Factor (GCF) (B)

▶️ Finding Least Common Multiple (LCM) (A)

🔒 Finding Least Common Multiple (LCM) (B)

▶️ Extended Practice: Finding GCF and LCM

▶️ Plotting Points (All Quadrants) (A)

🔒 Plotting Points (All Quadrants) (B)

▶️ Reading Points (All Quadrants) (A)

🔒 Reading Points (All Quadrants) (B)

▶️ Translations on the coordinate plane (A)

🔒 Translations on the coordinate plane (B)

▶️ Rotations on the coordinate plane (A)

🔒 Rotations on the coordinate plane (B)

▶️ Reflections on the coordinate plane (A)

🔒 Reflections on the coordinate plane (B)

▶️ Dilations on the coordinate plane (A)

🔒 Dilations on the coordinate plane (B)

▶️ Evaluating exponents (A)

🔒 Evaluating exponents (B)

▶️ Powers of 10 (A)

🔒 Powers of 10 (B)

▶️ Negative or zero exponents (A)

🔒 Negative or zero exponents (B)

▶️ Multiplying Exponents with the Same Base (A)

🔒 Multiplying Exponents with the Same Base (B)

▶️ Dividing Exponents with the Same Base (A)

🔒 Dividing Exponents with the Same Base (B)

▶️ Power to a Power (A)

🔒 Power to a Power (B)

▶️ Practice with Scientific Notation (A)

🔒 Practice with Scientific Notation (B)

🔒 Practice with Scientific Notation (C)

▶️ Extended Practice: Scientific Notation

▶️ Practice with square roots (A)

🔒 Practice with square roots (B)

🔒 Practice with square roots (C)

▶️ Estimating Square Roots of Non-Perfect Squares (A)

🔒 Estimating Square Roots of Non-Perfect Squares (B)

▶️ Simplifying Radicals (A)

🔒 Simplifying Radicals (B)

▶️ Adding and Subtracting Radicals (A)

🔒 Adding and Subtracting Radicals (B)

▶️ Multiplying and Dividing Radicals (A)

🔒 Multiplying and Dividing Radicals (B)

▶️ Practice with the Pythagorean Theorem (A)

▶️ Practice with the Pythagorean Theorem (B)

🔒 Practice with the Pythagorean Theorem (C)

🔒 Practice with the Pythagorean Theorem (D)

🔒 Practice with the Pythagorean Theorem (E)

▶️ Midpoint formula (A)

🔒 Midpoint formula (B)

▶️ Distance formula (A)

🔒 Distance formula (B)

▶️ Solving Radical Equations (A)

🔒 Solving Radical Equations (B)

▶️ Naming polynomials (A)

🔒 Naming polynomials (B)

▶️ Factoring monomials (A)

🔒 Factoring monomials (B)

▶️ Multiplying binomials (A)

🔒 Multiplying binomials (B)

▶️ Adding and subtracting polynomials (A)

🔒 Adding and subtracting polynomials (B)

▶️ Multiplying polynomials (A)

🔒 Multiplying polynomials (B)

▶️ Dividing polynomials (A)

🔒 Dividing polynomials (B)

▶️ Factoring the difference of two squares (A)

🔒 Factoring the difference of two squares (B)

▶️ Factoring trinomials when a=1 (A)

🔒 Factoring trinomials when a=1 (B)

▶️ Factoring trinomials when a ≠ 1 (A)

🔒 Factoring trinomials when a ≠ 1 (B)

▶️ Translating variable expressions into words (A)

🔒 Translating variable expressions into words (B)

▶️ Translating variable equations into words (A)

🔒 Translating variable equations into words (B)

▶️ Writing algebraic expressions (one step) (A)

🔒 Writing algebraic expressions (one step) (B)

🔒 Writing algebraic expressions (one step) (C)

▶️ Writing algebraic expressions (one or two step) (A)

🔒 Writing algebraic expressions (one or two step) (B)

🔒 Writing algebraic expressions (one or two step) (C)

▶️ Evaluating expressions (add/subtract) (one variable) (A)

🔒 Evaluating expressions (add/subtract) (one variable) (B)

▶️ Evaluating expressions (multiply/divide) (one variable) (A)

🔒 Evaluating expressions (multiply/divide) (one variable) (B)

▶️ Evaluating expressions (4 operations) (one variable) (A)

🔒 Evaluating expressions (4 operations) (one variable) (B)

▶️ Evaluating expressions (w/ exponents) (one variable) (A)

🔒 Evaluating expressions (w/ exponents) (one variable) (B)

▶️ Evaluating expressions (add/subtract) (two variables) (A)

🔒 Evaluating expressions (add/subtract) (two variables) (B)

▶️ Evaluating expressions (multiply/divide) (two variables) (A)

🔒 Evaluating expressions (multiply/divide) (two variables) (B)

▶️ Evaluating expressions (4 operations) (two variables) (A)

🔒 Evaluating expressions (4 operations) (two variables) (B)

▶️ Evaluating expressions (w/ exponents) (two variables) (A)

🔒 Evaluating expressions (w/ exponents) (two variables) (B)

▶️ Simplifying expressions (combine like terms) (A)

🔒 Simplifying expressions (combine like terms) (B)

🔒 Simplifying expressions (combine like terms) (C)

▶️ Solving one-step algebraic equations (add/subtract) (A)

🔒 Solving one-step algebraic equations (add/subtract) (B)

▶️ Solving one-step algebraic equations (multiply/divide) (A)

🔒 Solving one-step algebraic equations (multiply/divide) (B)

▶️ Solving one-step algebraic equations (4 operations) (A)

🔒 Solving one-step algebraic equations (4 operations) (B)

🔒 Solving one-step algebraic equations (4 operations) (C)

▶️ Solving two-step algebraic equations (A)

🔒 Solving two-step algebraic equations (B)

🔒 Solving two-step algebraic equations (C)

▶️ Solving 2-sided algebraic equations (A)

🔒 Solving 2-sided algebraic equations (B)

🔒 Solving 2-sided algebraic equations (C)

▶️ Completing two variable equation tables (A)

🔒 Completing two variable equation tables (B)

🔒 Completing two variable equation tables (C)

▶️ Writing equations using tables (A)

🔒 Writing equations using tables (B)

🔒 Writing equations using tables (C)

▶️ Solving multi-step equations (A)

▶️ Solving multi-step equations (B)

🔒 Solving multi-step equations (C)

▶️ Solving absolute value equations (A)

🔒 Solving absolute value equations (B)

▶️ Parallel lines and transversals (algebraic) (A)

🔒 Parallel lines and transversals (algebraic) (B)

▶️ Graphing single-variable inequalities on a number line (A)

🔒 Graphing single-variable inequalities on a number line (B)

▶️ Solving one-step inequalities (A)

🔒 Solving one-step inequalities (B)

▶️ Solving two-step inequalities (A)

🔒 Solving two-step inequalities (B)

▶️ Solving multi-step inequalities (A)

🔒 Solving multi-step inequalities (B)

▶️ Graphing linear inequalities on the coordinate plane (A)

🔒 Graphing linear inequalities on the coordinate plane (B)

🔒 Graphing linear inequalities on the coordinate plane (C)

▶️ Finding slope from a graph (A)

🔒 Finding slope from a graph (B)

▶️ Finding slope using a formula (A)

🔒 Finding slope using a formula (B)

▶️ Finding slope from an equation (A)

🔒 Finding slope from an equation (B)

▶️ Practice with Slope-Intercept Form (A)

🔒 Practice with Slope-Intercept Form (B)

🔒 Practice with Slope-Intercept Form (C)

▶️ Graphing lines in slope-intercept form (A)

▶️ Graphing lines in slope-intercept form (B)

🔒 Graphing lines in slope-intercept form (C)

▶️ Practice Writing Linear Equations (A)

🔒 Practice Writing Linear Equations (B)

▶️ Classifying parallel and perpendicular lines (A)

🔒 Classifying parallel and perpendicular lines (B)

▶️ Practice with Linear Inequalities on the Coordinate Plane (A)

▶️ Practice with Linear Inequalities on the Coordinate Plane (B)

🔒 Practice with Linear Inequalities on the Coordinate Plane (C)

🔒 Practice with Linear Inequalities on the Coordinate Plane (D)

🔒 Practice with Linear Inequalities on the Coordinate Plane (E)

▶️ Graphing absolute value functions (A)

🔒 Graphing absolute value functions (B)

🔒 Graphing absolute value functions (C)

▶️ Solving linear systems of equations by graphing (A)

▶️ Solving linear systems of equations by graphing (B)

🔒 Solving linear systems of equations by graphing (C)

🔒 Solving linear systems of equations by graphing (D)

🔒 Solving linear systems of equations by graphing (E)

▶️ Solving systems of linear inequalities by graphing (A)

▶️ Solving systems of linear inequalities by graphing (B)

🔒 Solving systems of linear inequalities by graphing (C)

🔒 Solving systems of linear inequalities by graphing (D)

🔒 Solving systems of linear inequalities by graphing (E)

▶️ Solving Quadratic Functions by Factoring (A)

🔒 Solving Quadratic Functions by Factoring (B)

🔒 Solving Quadratic Functions by Factoring (C)

▶️ Solving Quadratic Functions by Completing the Square (A)

🔒 Solving Quadratic Functions by Completing the Square (B)

🔒 Solving Quadratic Functions by Completing the Square (C )

▶️ Graphing Quadratic Functions (A)

🔒 Graphing Quadratic Functions (B)

🔒 Graphing Quadratic Functions (C)

▶️ Practice with Mean, Median, and Mode (A)

🔒 Practice with Mean, Median, and Mode (B)

▶️ Drawing line graphs (A)

🔒 Drawing line graphs (B)

▶️ Analyzing line graphs (A)

▶️ Analyzing line graphs (B)

🔒 Analyzing line graphs (C)

🔒 Analyzing line graphs (D)

▶️ Drawing Double Line Graphs (A)

🔒 Drawing Double Line Graphs (B)

▶️ Analyzing Double Line Graphs (A)

▶️ Analyzing Double Line Graphs (B)

🔒 Analyzing Double Line Graphs (C)

▶️ Making Box-and-Whisker Plots (A)

🔒 Making Box-and-Whisker Plots (B)

🔒 Making Box-and-Whisker Plots (C)

▶️ Analyzing Box-and-Whisker Plots (A)

🔒 Analyzing Box-and-Whisker Plots (B)

🔒 Analyzing Box-and-Whisker Plots (C)

▶️ Practice with Scatter Plots (A)

🔒 Practice with Scatter Plots (B)

▶️ Modeling situations using linear equations word problems (A)

🔒 Modeling situations using linear equations word problems (B)

▶️ Variables and expressions word problems (A)

🔒 Variables and expressions word problems (B)

🔒 Variables and expressions word problems (C)

▶️ Variables and equations word problems (A)

🔒 Variables and equations word problems (B)

🔒 Variables and equations word problems (C)

▶️ Scientific Notation Word Problems (A)

🔒 Scientific Notation Word Problems (B)

▶️ GCF and LCM Word Problems (A)

🔒 GCF and LCM Word Problems (B)

▶️ Ratio word problems (A)

🔒 Ratio word problems (B)

▶️ Proportion word problems (A)

🔒 Proportion word problems (B)

▶️ Pythagorean Theorem word problems (A)

🔒 Pythagorean Theorem word problems (B)

🔒 Pythagorean Theorem word problems (C)

▶️ Distance, rate, and time word problems (A)

🔒 Distance, rate, and time word problems (B)

▶️ Multi-step Word Problem: Snappy Rental Car

▶️ Multi-step Word Problem: Neil’s Square Paper

🔒 Multi-step Word Problem: Lilly Goes Shopping

🔒 Multi-step Word Problem: Pumpkins and Watermelons

🔒 Multi-step Word Problem: Alejandro’s Rock Collection

▶️ Percent Increase/Decrease Word Problems (A)

🔒 Percent Increase/Decrease Word Problems (B)

🛑 WAIT! Are You Looking to Unlock ALL of Our Algebra Worksheets?

When you sign up for the Mashup Math Infinite K-8 Worksheet Portal, you will gain on-demand access to our complete Algebra 1 Worksheets Library!

▶️ Click here to gain on-demand access to all of our Algebra Worksheets

Why Will Your Students will Love Our Algebra Worksheets?

It won’t take long for you to why Mashup Math Algebra Worksheets are the perfect supplement to your lesson plans and instruction.

Our worksheets can boost student engagement and participation because they are designed with the belief that effective math resources should never serve as busy work or focus on making students perform the same procedures mindlessly over and over again. Instead, our Algebra 1 worksheets were designed to help you give your students meaningful opportunities to practice and apply their way skills in a way that they will find interesting and appropriately challenging.

We are highly confident that our worksheets will be an asset to you and your lesson plans and that using them will help you to present math in a way that students will not see as repetitive or dull. Mashup Math worksheets make a great supplement to any lesson plan or homework assignment and having access to our complete Algebra Worksheets library will come in handy countless times throughout the school year.

Our Algebra 1 Worksheets can be used in a variety of ways including homework assignments, warm-up and cool-down activities, and topic reviews.

If you need suggestions our how the best use our Algebra 1 worksheets with your students, here are a few awesome ideas for ways that you can use our worksheets:

Students practice or study a topic independently

Students complete assignments for extra credit

Students complete worksheet problems during the first 5 minutes of class (warm-up)

Students complete worksheet problems during the last 5 minutes of class (exit ticket)

Students complete worksheets as homework assignments

Students complete worksheets to review previously learned topics

Formative assessments

Summative assessments

These ideas are all effective ways to use our algebra worksheets to supplement your algebra lessons and support your students. By giving your algebra students ample opportunities to engage with, explore, and practice math in an engaging way, you are supporting their development as mathematicians and problem solvers.

Great algebra teachers use effective and engaging topic-specific activities and resources to keep their lessons fresh and interesting for students, and our library of algebra 1 worksheets is the perfect addition to any teacher’s portfolio. When it comes to best meeting the needs of your students, it is crucial to present algebra in a way that is inspirational, interesting, and engaging in order create an environment where they are appropriately challenged and able to strengthen their math skills to levels that will allows them to be successful in college and beyond.

Do you want more Algebra Worksheets? Click here to gain access to our complete library of worksheets and answer keys , where you can gain on-demand access to ALL of our Algebra Worksheets.

• Get started
• Pre-Algebra

A quicker path to better grades

We have gathered all your curriculum-based courses, assignments, hints, tests, and solutions in one easy-to-use place

• Integrated I
• Integrated II
• Integrated III

Can't find your textbook?

More math. less studying.

A personal private tutor for each student. Free from preassure and study anxiety.

• Find your textbook
• Math Solver

• Big Ideas Math Algebra 1, 2015

• Big Ideas Math Algebra 1, 2013

• Big Ideas Math Algebra 1 Virginia

• Big Ideas Math Algebra 1 Texas

• Big Ideas Math Algebra 1 A Bridge to Success

• Core Connections Algebra 1, 2013

• Houghton Mifflin Harcourt Algebra 1, 2015

• Holt McDougal Algebra 1, 2011

• McDougal Littell Algebra 1, 1999

• McGraw Hill Glencoe Algebra 1, 2012

• McGraw Hill Glencoe Algebra 1, 2017

• McGraw Hill Glencoe Algebra 1 Texas, 2016

• Pearson Algebra 1 Common Core, 2011

• Pearson Algebra 1 Common Core, 2015

Algebra Nation works great on IE9, IE10, and the 3 most recent versions of Firefox, Chrome, and Safari.

Windows XP users should note that IE9 and IE10 will not run on Windows XP and therefore will need to use Firefox or Chrome. Windows XP users should also note that Windows XP will no longer be supported by Microsoft as of April 8th, 2014.

To get the best Algebra Nation 1.0 experience, we suggest you download the latest version of Google Chrome , Mozilla Firefox , or Internet Explorer .

Algebra Nation is Hiring

Algebra nation access.

As you may be aware, Math Nation did not receive state funding for the 2019-2020 school year. Individual districts and schools now purchase either workbooks (including digital access) or district-wide stand-alone digital access.

As of today, your school has not purchased at least 2 sets of workbooks with digital access or district-wide stand-alone digital access and so digital access will be limited on October 15. If your school or district purchased digital access for only 1 specific course, you will still continue to receive the unlimited version of that course, as well.

Please note that because of a gift from a private donor, the On-Ramp to Algebra 1 tool will continue to be available to your school and all schools in Florida at no cost during the 2019-2020 school year.

To learn more, or purchase workbooks or digital access, please email [email protected]

Alabama 3.0 Teacher Manager

Math Nation is launching a brand-new platform for the 2023-2024 school year that includes more resources, language supports, and accessibility features than ever before!

Starting May 26th, you will see a different look and feel when logging in. Please email

[email protected] for any questions and support.

Math Nation

Your School: If you can't find your school, click here . --> Questions? Don't see your school listed? Email [email protected] .

If you need help, drop us a line or give us a call. We'd love to hear from you!

Drawing Saved

We have a very quick question for you, so that we can make Math Nation better. Please answer honestly. This isn't a quiz or a test, and doesn't count for any sort of grade:

How much do you agree with the statement below?

Just a minute, please!

Send your feedback to the math nation team.

This is from submitting general feedback or to report an issue with the website. Your comments will be sent directly to the Math Nation team

Free Printable Math Worksheets for Algebra 1

Created with infinite algebra 1, stop searching. create the worksheets you need with infinite algebra 1..

• Fast and easy to use
• Multiple-choice & free-response
• Never runs out of questions
• Multiple-version printing

Free 14-Day Trial

• Writing variable expressions
• Order of operations
• Evaluating expressions
• Number sets
• Adding rational numbers
• Adding and subtracting rational numbers
• Multiplying and dividing rational numbers
• The distributive property
• Combining like terms
• Percent of change
• One-step equations
• Two-step equations
• Multi-step equations
• Absolute value equations
• Solving proportions
• Percent problems
• Distance-rate-time word problems
• Mixture word problems
• Work word problems
• Literal Equations
• Graphing one-variable inequalities
• One-step inequalities
• Two-step inequalities
• Multi-step inequalities
• Compound inequalities
• Absolute value inequalities
• Discrete relations
• Continuous relations
• Evaluating and graphing functions
• Finding slope from a graph
• Finding slope from two points
• Finding slope from an equation
• Graphing lines using slope-intercept form
• Graphing lines using standard form
• Writing linear equations
• Graphing linear inequalities
• Graphing absolute value equations
• Direct variation
• Solving systems of equations by graphing
• Solving systems of equations by elimination
• Solving systems of equations by substitution
• Systems of equations word problems
• Graphing systems of inequalities
• Discrete exponential growth and decay word problems
• Exponential functions and graphs
• Writing numbers in scientific notation
• Operations with scientific notation
• Addition and subtraction with scientific notation
• Naming polynomials
• Adding and subtracting polynomials
• Multiplying polynomials
• Multiplying special case polynomials
• Factoring special case polynomials
• Factoring by grouping
• Dividing polynomials
• Graphing quadratic inequalities
• Completing the square
• By taking square roots
• By factoring
• With the quadratic formula
• By completing the square
• Simplifying radicals
• Adding and subtracting radical expressions
• Multiplying radicals
• Dividing radicals
• Using the distance formula
• Using the midpoint formula
• Simplifying rational expressions
• Finding excluded values / restricted values
• Multiplying rational expressions
• Dividing rational expressions
• Adding and subtracting rational expressions
• Finding trig. ratios
• Finding angles of triangles
• Finding side lengths of triangles
• Visualizing data
• Center and spread of data
• Scatter plots
• Using statistical models

• school Campus Bookshelves
• menu_book Bookshelves
• perm_media Learning Objects
• login Login
• how_to_reg Request Instructor Account
• hub Instructor Commons

Margin Size

• Download Page (PDF)
• Download Full Book (PDF)
• Periodic Table
• Physics Constants
• Scientific Calculator
• Reference & Cite
• Tools expand_more
• Readability

selected template will load here

This action is not available.

3.2: Functions and Function Notation

• Last updated
• Save as PDF
• Page ID 15053

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)

( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)

\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)

\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

\( \newcommand{\Span}{\mathrm{span}}\)

\( \newcommand{\id}{\mathrm{id}}\)

\( \newcommand{\kernel}{\mathrm{null}\,}\)

\( \newcommand{\range}{\mathrm{range}\,}\)

\( \newcommand{\RealPart}{\mathrm{Re}}\)

\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

\( \newcommand{\Argument}{\mathrm{Arg}}\)

\( \newcommand{\norm}[1]{\| #1 \|}\)

\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)

\( \newcommand{\vectorA}[1]{\vec{#1}}      % arrow\)

\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}}      % arrow\)

\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

\( \newcommand{\vectorC}[1]{\textbf{#1}} \)

\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)

\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)

\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)

Learning Objectives

• Determine whether a relation represents a function.
• Find the value of a function.
• Determine whether a function is one-to-one.
• Use the vertical line test to identify functions.
• Graph the functions listed in the library of functions.

A jetliner changes altitude as its distance from the starting point of a flight increases. The weight of a growing child increases with time. In each case, one quantity depends on another. There is a relationship between the two quantities that we can describe, analyze, and use to make predictions. In this section, we will analyze such relationships.

Determining Whether a Relation Represents a Function

A relation is a set of ordered pairs. The set of the first components of each ordered pair is called the domain and the set of the second components of each ordered pair is called the range. Consider the following set of ordered pairs. The first numbers in each pair are the first five natural numbers. The second number in each pair is twice that of the first.

\[\{(1, 2), (2, 4), (3, 6), (4, 8), (5, 10)\}\tag{1.1.1}\]

The domain is \(\{1, 2, 3, 4, 5\}\). The range is \(\{2, 4, 6, 8, 10\}\).

Note that each value in the domain is also known as an input value, or independent variable , and is often labeled with the lowercase letter \(x\). Each value in the range is also known as an output value, or dependent variable , and is often labeled lowercase letter \(y\).

A function \(f\) is a relation that assigns a single value in the range to each value in the domain. In other words, no \(x\)-values are repeated. For our example that relates the first five natural numbers to numbers double their values, this relation is a function because each element in the domain, {1, 2, 3, 4, 5}, is paired with exactly one element in the range, \(\{2, 4, 6, 8, 10\}\).

Now let’s consider the set of ordered pairs that relates the terms “even” and “odd” to the first five natural numbers. It would appear as

\[\mathrm{\{(odd, 1), (even, 2), (odd, 3), (even, 4), (odd, 5)\}} \tag{1.1.2}\]

Notice that each element in the domain, {even, odd} is not paired with exactly one element in the range, \(\{1, 2, 3, 4, 5\}\). For example, the term “odd” corresponds to three values from the range, \(\{1, 3, 5\},\) and the term “even” corresponds to two values from the range, \(\{2, 4\}\). This violates the definition of a function, so this relation is not a function.

Figure \(\PageIndex{1}\) compares relations that are functions and not functions.

A function is a relation in which each possible input value leads to exactly one output value. We say “the output is a function of the input.”

The input values make up the domain , and the output values make up the range .

How To: Given a relationship between two quantities, determine whether the relationship is a function

• Identify the input values.
• Identify the output values.
• If each input value leads to only one output value, classify the relationship as a function. If any input value leads to two or more outputs, do not classify the relationship as a function.

Example \(\PageIndex{1}\): Determining If Menu Price Lists Are Functions

The coffee shop menu, shown in Figure \(\PageIndex{2}\) consists of items and their prices.

• Is price a function of the item?
• Is the item a function of the price?

• Let’s begin by considering the input as the items on the menu. The output values are then the prices. See Figure \(\PageIndex{3}\).

Each item on the menu has only one price, so the price is a function of the item.

• Two items on the menu have the same price. If we consider the prices to be the input values and the items to be the output, then the same input value could have more than one output associated with it. See Figure \(\PageIndex{4}\).

Therefore, the item is a not a function of price.

Example \(\PageIndex{2}\): Determining If Class Grade Rules Are Functions

In a particular math class, the overall percent grade corresponds to a grade point average. Is grade point average a function of the percent grade? Is the percent grade a function of the grade point average? Table \(\PageIndex{1}\) shows a possible rule for assigning grade points.

For any percent grade earned, there is an associated grade point average, so the grade point average is a function of the percent grade. In other words, if we input the percent grade, the output is a specific grade point average.

In the grading system given, there is a range of percent grades that correspond to the same grade point average. For example, students who receive a grade point average of 3.0 could have a variety of percent grades ranging from 78 all the way to 86. Thus, percent grade is not a function of grade point average.

Exercise \(\PageIndex{2}\)

Table \(\PageIndex{2}\) lists the five greatest baseball players of all time in order of rank.

• Is the rank a function of the player name?
• Is the player name a function of the rank?

yes. (Note: If two players had been tied for, say, 4th place, then the name would not have been a function of rank.)

Using Function Notation

Once we determine that a relationship is a function, we need to display and define the functional relationships so that we can understand and use them, and sometimes also so that we can program them into computers. There are various ways of representing functions. A standard function notation is one representation that facilitates working with functions.

To represent “height is a function of age,” we start by identifying the descriptive variables \(h\) for height and \(a\) for age. The letters \(f\), \(g\),and \(h\) are often used to represent functions just as we use \(x\), \(y\),and \(z\) to represent numbers and \(A\), \(B\), and \(C\) to represent sets.

\[\begin{array}{ll} h \text{ is } f \text{ of }a \;\;\;\;\;\; & \text{We name the function }f \text{; height is a function of age.} \\ h=f(a) & \text{We use parentheses to indicate the function input.} \\ f(a) & \text{We name the function }f \text{ ; the expression is read as “ }f \text{ of }a \text{.”}\end{array}\]

Remember, we can use any letter to name the function; the notation \(h(a)\) shows us that \(h\) depends on \(a\). The value \(a\) must be put into the function \(h\) to get a result. The parentheses indicate that age is input into the function; they do not indicate multiplication.

We can also give an algebraic expression as the input to a function. For example \(f(a+b)\) means “first add \(a\) and \(b\), and the result is the input for the function \(f\).” The operations must be performed in this order to obtain the correct result.

Function Notation

The notation \(y=f(x)\) defines a function named \(f\). This is read as “\(y\) is a function of \(x\).” The letter \(x\) represents the input value, or independent variable. The letter \(y\), or \(f(x)\), represents the output value, or dependent variable.

Example \(\PageIndex{3}\): Using Function Notation for Days in a Month

Use function notation to represent a function whose input is the name of a month and output is the number of days in that month.

Using Function Notation for Days in a Month

The number of days in a month is a function of the name of the month, so if we name the function \(f\), we write \(\text{days}=f(\text{month})\) or \(d=f(m)\). The name of the month is the input to a “rule” that associates a specific number (the output) with each input.

For example, \(f(\text{March})=31\), because March has 31 days. The notation \(d=f(m)\) reminds us that the number of days, \(d\) (the output), is dependent on the name of the month, \(m\) (the input).

Note that the inputs to a function do not have to be numbers; function inputs can be names of people, labels of geometric objects, or any other element that determines some kind of output. However, most of the functions we will work with in this book will have numbers as inputs and outputs.

Example \(\PageIndex{3B}\): Interpreting Function Notation

A function \(N=f(y)\) gives the number of police officers, \(N\), in a town in year \(y\). What does \(f(2005)=300\) represent?

When we read \(f(2005)=300\), we see that the input year is 2005. The value for the output, the number of police officers \((N)\), is 300. Remember, \(N=f(y)\). The statement \(f(2005)=300\) tells us that in the year 2005 there were 300 police officers in the town.

Exercise \(\PageIndex{3}\)

Use function notation to express the weight of a pig in pounds as a function of its age in days \(d\).

Instead of a notation such as \(y=f(x)\), could we use the same symbol for the output as for the function, such as \(y=y(x)\), meaning “\(y\) is a function of \(x\)?”

Yes, this is often done, especially in applied subjects that use higher math, such as physics and engineering. However, in exploring math itself we like to maintain a distinction between a function such as \(f\) , which is a rule or procedure, and the output y we get by applying \(f\) to a particular input \(x\) . This is why we usually use notation such as \(y=f(x),P=W(d)\) , and so on.

Representing Functions Using Tables

A common method of representing functions is in the form of a table. The table rows or columns display the corresponding input and output values. In some cases, these values represent all we know about the relationship; other times, the table provides a few select examples from a more complete relationship.

Table \(\PageIndex{3}\) lists the input number of each month (\(\text{January}=1\), \(\text{February}=2\), and so on) and the output value of the number of days in that month. This information represents all we know about the months and days for a given year (that is not a leap year). Note that, in this table, we define a days-in-a-month function \(f\) where \(D=f(m)\) identifies months by an integer rather than by name.

Table \(\PageIndex{4}\) defines a function \(Q=g(n)\) Remember, this notation tells us that \(g\) is the name of the function that takes the input \(n\) and gives the output \(Q\).

Table \(\PageIndex{5}\) displays the age of children in years and their corresponding heights. This table displays just some of the data available for the heights and ages of children. We can see right away that this table does not represent a function because the same input value, 5 years, has two different output values, 40 in. and 42 in.

How To: Given a table of input and output values, determine whether the table represents a function

• Identify the input and output values.
• Check to see if each input value is paired with only one output value. If so, the table represents a function.

Example \(\PageIndex{5}\): Identifying Tables that Represent Functions

Which table, Table \(\PageIndex{6}\), Table \(\PageIndex{7}\), or Table \(\PageIndex{8}\), represents a function (if any)?

Table \(\PageIndex{6}\) and Table \(\PageIndex{7}\) define functions. In both, each input value corresponds to exactly one output value. Table \(\PageIndex{8}\) does not define a function because the input value of 5 corresponds to two different output values.

When a table represents a function, corresponding input and output values can also be specified using function notation.

The function represented by Table \(\PageIndex{6}\) can be represented by writing

\[f(2)=1\text{, }f(5)=3\text{, and }f(8)=6 \nonumber\]

Similarly, the statements

\[g(−3)=5\text{, }g(0)=1\text{, and }g(4)=5 \nonumber\]

represent the function in Table \(\PageIndex{7}\).

Table \(\PageIndex{8}\) cannot be expressed in a similar way because it does not represent a function.

Exercise \(\PageIndex{5}\)

Does Table \(\PageIndex{9}\) represent a function?

Finding Input and Output Values of a Function

When we know an input value and want to determine the corresponding output value for a function, we evaluate the function. Evaluating will always produce one result because each input value of a function corresponds to exactly one output value.

When we know an output value and want to determine the input values that would produce that output value, we set the output equal to the function’s formula and solve for the input. Solving can produce more than one solution because different input values can produce the same output value.

Evaluation of Functions in Algebraic Forms

When we have a function in formula form, it is usually a simple matter to evaluate the function. For example, the function \(f(x)=5−3x^2\) can be evaluated by squaring the input value, multiplying by 3, and then subtracting the product from 5.

How To: Given the formula for a function, evaluate.

Given the formula for a function, evaluate.

• Replace the input variable in the formula with the value provided.
• Calculate the result.

Example \(\PageIndex{6A}\): Evaluating Functions at Specific Values

1. Evaluate \(f(x)=x^2+3x−4\) at

• Evaluate \(\frac{f(a+h)−f(a)}{h}\)

Replace the x in the function with each specified value.

a. Because the input value is a number, 2, we can use simple algebra to simplify.

\[\begin{align*}f(2)&=2^2+3(2)−4\\&=4+6−4\\ &=6\end{align*}\]

b. In this case, the input value is a letter so we cannot simplify the answer any further.

\[f(a)=a^2+3a−4\nonumber\]

c. With an input value of \(a+h\), we must use the distributive property.

\[\begin{align*}f(a+h)&=(a+h)^2+3(a+h)−4\\&=a^2+2ah+h^2+3a+3h−4 \end{align*}\]

d. In this case, we apply the input values to the function more than once, and then perform algebraic operations on the result. We already found that

\[f(a+h)=a^2+2ah+h^2+3a+3h−4\nonumber\]

and we know that

\[f(a)=a^2+3a−4 \nonumber\]

Now we combine the results and simplify.

\[\begin{align*}\dfrac{f(a+h)−f(a)}{h}&=\dfrac{(a^2+2ah+h^2+3a+3h−4)−(a^2+3a−4)}{h}\\ &=\dfrac{(2ah+h^2+3h)}{h} \\ &=\dfrac{h(2a+h+3)}{h} & &\text{Factor out h.}\\ &=2a+h+3 & & \text{Simplify.}\end{align*}\]

Example \(\PageIndex{6B}\): Evaluating Functions

Given the function \(h(p)=p^2+2p\), evaluate \(h(4)\).

To evaluate \(h(4)\), we substitute the value 4 for the input variable p in the given function.

\[\begin{align*}h(p)&=p^2+2p\\h(4)&=(4)^2+2(4)\\ &=16+8\\&=24\end{align*}\]

Therefore, for an input of 4, we have an output of 24.

Exercise \(\PageIndex{6}\)

Given the function \(g(m)=\sqrt{m−4}\), evaluate \(g(5)\).

Example \(\PageIndex{7}\): Solving Functions

Given the function \(h(p)=p^2+2p\), solve for \(h(p)=3\).

\[\begin{array}{rl} h(p)=3\\p^2+2p=3 & \text{Substitute the original function}\\ p^2+2p−3=0 & \text{Subtract 3 from each side.}\\(p+3)(p−1)=0&\text{Factor.}\end{array} \nonumber \]

If \((p+3)(p−1)=0\), either \((p+3)=0\) or \((p−1)=0\) (or both of them equal \(0\)). We will set each factor equal to \(0\) and solve for \(p\) in each case.

\[(p+3)=0,\; p=−3 \nonumber \]

\[(p−1)=0,\, p=1 \nonumber\]

This gives us two solutions. The output \(h(p)=3\) when the input is either \(p=1\) or \(p=−3\). We can also verify by graphing as in Figure \(\PageIndex{6}\). The graph verifies that \(h(1)=h(−3)=3\) and \(h(4)=24\).

Exercise \(\PageIndex{7}\)

Given the function \(g(m)=\sqrt{m−4}\), solve \(g(m)=2\).

Evaluating Functions Expressed in Formulas

Some functions are defined by mathematical rules or procedures expressed in equation form. If it is possible to express the function output with a formula involving the input quantity, then we can define a function in algebraic form. For example, the equation \(2n+6p=12\) expresses a functional relationship between \(n\) and \(p\). We can rewrite it to decide if \(p\) is a function of \(n\).

How to: Given a function in equation form, write its algebraic formula.

• Solve the equation to isolate the output variable on one side of the equal sign, with the other side as an expression that involves only the input variable.
• Use all the usual algebraic methods for solving equations, such as adding or subtracting the same quantity to or from both sides, or multiplying or dividing both sides of the equation by the same quantity.

Example \(\PageIndex{8A}\): Finding an Equation of a Function

Express the relationship \(2n+6p=12\) as a function \(p=f(n)\), if possible.

To express the relationship in this form, we need to be able to write the relationship where \(p\) is a function of \(n\), which means writing it as \(p=[\text{expression involving }n]\).

\[\begin{align*}2n+6p&=12 \\ 6p&=12−2n && \text{Subtract 2n from both sides.} \\ p&=\dfrac{12−2n}{6} & &\text{Divide both sides by 6 and simplify.} \\ p&=\frac{12}{6}−\frac{2n}{6} \\ p&=2−\frac{1}{3}n\end{align*}\]

Therefore, \(p\) as a function of \(n\) is written as

\[p=f(n)=2−\frac{1}{3}n \nonumber\]

It is important to note that not every relationship expressed by an equation can also be expressed as a function with a formula.

Example \(\PageIndex{8B}\): Expressing the Equation of a Circle as a Function

Does the equation \(x^2+y^2=1\) represent a function with \(x\) as input and \(y\) as output? If so, express the relationship as a function \(y=f(x)\).

First we subtract \(x^2\) from both sides.

\[y^2=1−x^2 \nonumber\]

We now try to solve for \(y\) in this equation.

\[y=\pm\sqrt{1−x^2} \nonumber\]

\[\text{so, }y=\sqrt{1−x^2}\;\text{and}\;y = −\sqrt{1−x^2} \nonumber\]

We get two outputs corresponding to the same input, so this relationship cannot be represented as a single function \(y=f(x)\).

Exercise \(\PageIndex{8}\)

If \(x−8y^3=0\), express \(y\) as a function of \(x\).

\(y=f(x)=\dfrac{\sqrt[3]{x}}{2}\)

Are there relationships expressed by an equation that do represent a function but which still cannot be represented by an algebraic formula?

Yes, this can happen. For example, given the equation \(x=y+2^y\), if we want to express y as a function of x, there is no simple algebraic formula involving only \(x\) that equals \(y\). However, each \(x\) does determine a unique value for \(y\), and there are mathematical procedures by which \(y\) can be found to any desired accuracy. In this case, we say that the equation gives an implicit (implied) rule for \(y\) as a function of \(x\), even though the formula cannot be written explicitly.

Evaluating a Function Given in Tabular Form

As we saw above, we can represent functions in tables. Conversely, we can use information in tables to write functions, and we can evaluate functions using the tables. For example, how well do our pets recall the fond memories we share with them? There is an urban legend that a goldfish has a memory of 3 seconds, but this is just a myth. Goldfish can remember up to 3 months, while the beta fish has a memory of up to 5 months. And while a puppy’s memory span is no longer than 30 seconds, the adult dog can remember for 5 minutes. This is meager compared to a cat, whose memory span lasts for 16 hours.

The function that relates the type of pet to the duration of its memory span is more easily visualized with the use of a table (Table \(\PageIndex{10}\)).

At times, evaluating a function in table form may be more useful than using equations. Here let us call the function \(P\). The domain of the function is the type of pet and the range is a real number representing the number of hours the pet’s memory span lasts. We can evaluate the function \(P\) at the input value of “goldfish.” We would write \(P(goldfish)=2160\). Notice that, to evaluate the function in table form, we identify the input value and the corresponding output value from the pertinent row of the table. The tabular form for function P seems ideally suited to this function, more so than writing it in paragraph or function form.

How To: Given a function represented by a table, identify specific output and input values

1. Find the given input in the row (or column) of input values. 2. Identify the corresponding output value paired with that input value. 3. Find the given output values in the row (or column) of output values, noting every time that output value appears. 4. Identify the input value(s) corresponding to the given output value.

Example \(\PageIndex{9}\): Evaluating and Solving a Tabular Function

Using Table \(\PageIndex{11}\),

a. Evaluate \(g(3)\). b. Solve \(g(n)=6\).

a. Evaluating \(g(3)\) means determining the output value of the function \(g\) for the input value of \(n=3\). The table output value corresponding to \(n=3\) is 7, so \(g(3)=7\). b. Solving \(g(n)=6\) means identifying the input values, n,that produce an output value of 6. Table \(\PageIndex{12}\) shows two solutions: 2 and 4.

When we input 2 into the function \(g\), our output is 6. When we input 4 into the function \(g\), our output is also 6.

Exercise \(\PageIndex{1}\)

Using Table \(\PageIndex{12}\), evaluate \(g(1)\).

Finding Function Values from a Graph

Evaluating a function using a graph also requires finding the corresponding output value for a given input value, only in this case, we find the output value by looking at the graph. Solving a function equation using a graph requires finding all instances of the given output value on the graph and observing the corresponding input value(s).

Example \(\PageIndex{10}\): Reading Function Values from a Graph

Given the graph in Figure \(\PageIndex{7}\),

• Evaluate \(f(2)\).
• Solve \(f(x)=4\).

To evaluate \(f(2)\), locate the point on the curve where \(x=2\), then read the y-coordinate of that point. The point has coordinates \((2,1)\), so \(f(2)=1\). See Figure \(\PageIndex{8}\).

To solve \(f(x)=4\), we find the output value 4 on the vertical axis. Moving horizontally along the line \(y=4\), we locate two points of the curve with output value 4: \((−1,4)\) and \((3,4)\). These points represent the two solutions to \(f(x)=4\): −1 or 3. This means \(f(−1)=4\) and \(f(3)=4\), or when the input is −1 or 3, the output is 4. See Figure \(\PageIndex{9}\).

Exercise \(\PageIndex{10}\)

Given the graph in Figure \(\PageIndex{7}\), solve \(f(x)=1\).

\(x=0\) or \(x=2\)

Determining Whether a Function is One-to-One

Some functions have a given output value that corresponds to two or more input values. For example, in the stock chart shown in the Figure at the beginning of this chapter, the stock price was $1000 on five different dates, meaning that there were five different input values that all resulted in the same output value of $1000.

However, some functions have only one input value for each output value, as well as having only one output for each input. We call these functions one-to-one functions. As an example, consider a school that uses only letter grades and decimal equivalents, as listed in Table \(\PageIndex{13}\).

This grading system represents a one-to-one function, because each letter input yields one particular grade point average output and each grade point average corresponds to one input letter.

To visualize this concept, let’s look again at the two simple functions sketched in Figures \(\PageIndex{1a}\) and \(\PageIndex{1b}\). The function in part (a) shows a relationship that is not a one-to-one function because inputs \(q\) and \(r\) both give output \(n\). The function in part (b) shows a relationship that is a one-to-one function because each input is associated with a single output.

One-to-One Functions

A one-to-one function is a function in which each output value corresponds to exactly one input value.

Example \(\PageIndex{11}\): Determining Whether a Relationship Is a One-to-One Function

Is the area of a circle a function of its radius? If yes, is the function one-to-one?

A circle of radius \(r\) has a unique area measure given by \(A={\pi}r^2\), so for any input, \(r\), there is only one output, \(A\). The area is a function of radius\(r\).

If the function is one-to-one, the output value, the area, must correspond to a unique input value, the radius. Any area measure \(A\) is given by the formula \(A={\pi}r^2\). Because areas and radii are positive numbers, there is exactly one solution:\(\sqrt{\frac{A}{\pi}}\). So the area of a circle is a one-to-one function of the circle’s radius.

Exercise \(\PageIndex{11A}\)

• Is a balance a function of the bank account number?
• Is a bank account number a function of the balance?
• Is a balance a one-to-one function of the bank account number?

a. yes, because each bank account has a single balance at any given time;

b. no, because several bank account numbers may have the same balance;

c. no, because the same output may correspond to more than one input.

Exercise \(\PageIndex{11B}\)

Evaluate the following:

• If each percent grade earned in a course translates to one letter grade, is the letter grade a function of the percent grade?
• If so, is the function one-to-one?

a. Yes, letter grade is a function of percent grade; b. No, it is not one-to-one. There are 100 different percent numbers we could get but only about five possible letter grades, so there cannot be only one percent number that corresponds to each letter grade.

Using the Vertical Line Test

As we have seen in some examples above, we can represent a function using a graph. Graphs display a great many input-output pairs in a small space. The visual information they provide often makes relationships easier to understand. By convention, graphs are typically constructed with the input values along the horizontal axis and the output values along the vertical axis.

The most common graphs name the input value \(x\) and the output \(y\), and we say \(y\) is a function of \(x\), or \(y=f(x)\) when the function is named \(f\). The graph of the function is the set of all points \((x,y)\) in the plane that satisfies the equation \(y=f(x)\). If the function is defined for only a few input values, then the graph of the function is only a few points, where the x-coordinate of each point is an input value and the y-coordinate of each point is the corresponding output value. For example, the black dots on the graph in Figure \(\PageIndex{10}\) tell us that \(f(0)=2\) and \(f(6)=1\). However, the set of all points \((x,y)\) satisfying \(y=f(x)\) is a curve. The curve shown includes \((0,2)\) and \((6,1)\) because the curve passes through those points

The vertical line test can be used to determine whether a graph represents a function. If we can draw any vertical line that intersects a graph more than once, then the graph does not define a function because a function has only one output value for each input value. See Figure \(\PageIndex{11}\) .

Howto: Given a graph, use the vertical line test to determine if the graph represents a function

• Inspect the graph to see if any vertical line drawn would intersect the curve more than once.
• If there is any such line, determine that the graph does not represent a function.

Example \(\PageIndex{12}\): Applying the Vertical Line Test

Which of the graphs in Figure \(\PageIndex{12}\) represent(s) a function \(y=f(x)\)?

If any vertical line intersects a graph more than once, the relation represented by the graph is not a function. Notice that any vertical line would pass through only one point of the two graphs shown in parts (a) and (b) of Figure \(\PageIndex{12}\). From this we can conclude that these two graphs represent functions. The third graph does not represent a function because, at most x-values, a vertical line would intersect the graph at more than one point, as shown in Figure \(\PageIndex{13}\).

Exercise \(\PageIndex{12}\)

Does the graph in Figure \(\PageIndex{14}\) represent a function?

Using the Horizontal Line Test

Once we have determined that a graph defines a function, an easy way to determine if it is a one-to-one function is to use the horizontal line test . Draw horizontal lines through the graph. If any horizontal line intersects the graph more than once, then the graph does not represent a one-to-one function.

Howto: Given a graph of a function, use the horizontal line test to determine if the graph represents a one-to-one function

• Inspect the graph to see if any horizontal line drawn would intersect the curve more than once.
• If there is any such line, determine that the function is not one-to-one.

Example \(\PageIndex{13}\): Applying the Horizontal Line Test

Consider the functions shown in Figure \(\PageIndex{12a}\) and Figure \(\PageIndex{12b}\). Are either of the functions one-to-one?

The function in Figure \(\PageIndex{12a}\) is not one-to-one. The horizontal line shown in Figure \(\PageIndex{15}\) intersects the graph of the function at two points (and we can even find horizontal lines that intersect it at three points.)

The function in Figure \(\PageIndex{12b}\) is one-to-one. Any horizontal line will intersect a diagonal line at most once.

Exercise \(\PageIndex{13}\)

Is the graph shown in Figure \(\PageIndex{13}\) one-to-one?

No, because it does not pass the horizontal line test.

Identifying Basic Toolkit Functions

In this text, we will be exploring functions—the shapes of their graphs, their unique characteristics, their algebraic formulas, and how to solve problems with them. When learning to read, we start with the alphabet. When learning to do arithmetic, we start with numbers. When working with functions, it is similarly helpful to have a base set of building-block elements. We call these our “toolkit functions,” which form a set of basic named functions for which we know the graph, formula, and special properties. Some of these functions are programmed to individual buttons on many calculators. For these definitions we will use x as the input variable and \(y=f(x)\) as the output variable.

We will see these toolkit functions, combinations of toolkit functions, their graphs, and their transformations frequently throughout this book. It will be very helpful if we can recognize these toolkit functions and their features quickly by name, formula, graph, and basic table properties. The graphs and sample table values are included with each function shown in Table \(\PageIndex{14}\).

Key Equations

• Constant function \(f(x)=c\), where \(c\) is a constant
• Identity function \(f(x)=x\)
• Absolute value function \(f(x)=|x|\)
• Quadratic function \(f(x)=x^2\)
• Cubic function \(f(x)=x^3\)
• Reciprocal function \(f(x)=\dfrac{1}{x}\)
• Reciprocal squared function \(f(x)=\frac{1}{x^2}\)
• Square root function \(f(x)=\sqrt{x}\)
• Cube root function \(f(x)=3\sqrt{x}\)

Key Concepts

• A relation is a set of ordered pairs. A function is a specific type of relation in which each domain value, or input, leads to exactly one range value, or output.
• Function notation is a shorthand method for relating the input to the output in the form \(y=f(x)\).
• In tabular form, a function can be represented by rows or columns that relate to input and output values.
• To evaluate a function, we determine an output value for a corresponding input value. Algebraic forms of a function can be evaluated by replacing the input variable with a given value.
• To solve for a specific function value, we determine the input values that yield the specific output value.
• An algebraic form of a function can be written from an equation.
• Input and output values of a function can be identified from a table.
• Relating input values to output values on a graph is another way to evaluate a function.
• A function is one-to-one if each output value corresponds to only one input value.
• A graph represents a function if any vertical line drawn on the graph intersects the graph at no more than one point.
• The graph of a one-to-one function passes the horizontal line test.

1 http://www.baseball-almanac.com/lege.../lisn100.shtml . Accessed 3/24/2014. 2 www.kgbanswers.com/how-long-i...y-span/4221590. Accessed 3/24/2014.

dependent variable an output variable

domain the set of all possible input values for a relation

function a relation in which each input value yields a unique output value

horizontal line test a method of testing whether a function is one-to-one by determining whether any horizontal line intersects the graph more than once

independent variable an input variable

input each object or value in a domain that relates to another object or value by a relationship known as a function

one-to-one function a function for which each value of the output is associated with a unique input value

output each object or value in the range that is produced when an input value is entered into a function

range the set of output values that result from the input values in a relation

relation a set of ordered pairs

vertical line test a method of testing whether a graph represents a function by determining whether a vertical line intersects the graph no more than once

Please ensure that your password is at least 8 characters and contains each of the following:

• a special character: @$#!%*?&

Florida’s B.E.S.T. Curriculum by Math Nation

6th Grade Math – Volume 1 of 2

6th Grade Math – Volume 2 of 2

7th Grade Math – Volume 1 of 2

7th Grade Math – Volume 2 of 2

8th Grade Math – Volume 1 of 2

8th Grade Math – Volume 2 of 2

6th Grade Accelerated Math – Volume 1 of 2

6th Grade Accelerated Math – Volume 2 of 2

7th Grade Accelerated Math – Volume 1 of 2

7th Grade Accelerated Math – Volume 2 of 2

Algebra 1 – Volume 1 of 2

Algebra 1 – Volume 2 of 2

Geometry – Volume 1 of 2

Geometry – Volume 2 of 2

Algebra 2 – Volume 1 of 2

Algebra 2 – Volume 2 of 2

Which school do you go to?

If your school has access to Math Nation, please start typing and select your school below!

No school found.

Please email

(you can also download the free mobile app by searching "Math Nation" in your phone/tablet’s app store)

One More Thing!

To log in to Math Nation, please enter your Student ID Number and your birthdate.

Need help signing in? Please email [email protected] or call 1-888-608-MATH.

Are you a student or a teacher/administrator?

Welcome to math nation.

Math Nation is a dynamic free math resource for that provides dynamic instructional videos, workbooks, and interactive tutoring to students and teachers in New York City. Math Nation is 100% aligned to the New York Common Core Learning Standards. Setting up access is easy and free! Click here to learn more

If your school does not have access to Math Nation yet, don't worry! You can explore our resources right away.

Teacher Guest Login

Please enter your name and email to log in as a guest.

Log in as a Guest

You can log in as a guest or go back to try again.

If you have a question about logging in, please email [email protected]

Choose your state

• Mississippi
• New York City
• South Carolina

Choose your location

• Parent & Family FAQs
• Teacher FAQs
• Student FAQs

To continue, please click on the option that best describes you.

To continue, please download our app

Not on a mobile device? Login here

• Connecticut
• Massachusetts
• New Hampshire
• North Carolina
• North Dakota
• Northern Mariana Island
• Pennsylvania
• Rhode Island
• South Dakota
• West Virginia