introduction to functions assignment

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On this page

• Introduction to Geometry
• Functions and Graphs

1. Introduction to Functions

• 2. Functions from Verbal Statements
• 3. Rectangular Coordinates
• 4. The Graph of a Function
• 4a. Domain and Range of a Function
• 4b. Domain and Range interactive applet
• 4c. Comparison calculator BMI - BAI
• 5. Graphing Using a Computer Algebra System
• 5a. Online graphing calculator (1): Plot your own graph (JSXGraph)
• 5b. Online graphing calculator (2): Plot your own graph (SVG)
• 6. Graphs of Functions Defined by Tables of Data
• 7. Continuous and Discontinuous Functions
• 8. Split Functions
• 9. Even and Odd Functions

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In everyday life, many quantities depend on one or more changing variables. For example:

(a) Plant growth depends on sunlight and rainfall

(b) Speed depends on distance travelled and time taken

(c) Voltage depends on current and resistance

(d) Test marks depend on attitude, listening in lectures and doing tutorials (among many other variables!!)

A function is a rule that relates how one quantity depends on other quantities.

A particular electrical circuit has a power source and an 8 ohms (Ω) resistor. The voltage in that circult is given by:

V = voltage (in volts, V) I = current (in amperes, A)

So if I = 4 amperes, then the voltage is V = 8 × 4 = 32 volts.

If I increases, so does the voltage, V .

If I decreases, so does the voltage, V .

We say voltage is a function of current (when resistance is constant). We get only one value of V for each value of I .

A bicycle covers a distance in 20 seconds. The speed of the bicycle is given by

`s=d/20=0.05d`

s = speed (in ms −1 , or meters per second, m/s) d = distance (in meters, m)

If the distance covered by the bike is 10 m, then the speed is `s = 0.05 × 10 = 0.5\ "m/s"`.

If d increases, the speed goes up .

If d decreases, the speed goes down .

We say speed is a function of distance (when time is constant). We get only one value of s for each value of d .

Definition of a Function

We have 2 quantities (called "variables") and we observe there is a relationship between them. If we find that for every value of the first variable there is only one value of the second variable, then we say:

"The second variable is a function of the first variable."

The first variable is the independent variable (usually written as x ), and the second variable is the dependent variable (usually written as y ).

The independent variable and the dependent variable are real numbers . (We'll learn about numbers which are not real later, in Complex Numbers .)

We know the equation for the area, A , of a circle from primary school:

A = πr 2 , where r is the radius of the circle

This is a function as each value of the independent variable r gives us one value of the dependent variable A .

General Cases

We use x for the independent variable and y for the dependent variable for general cases. This is very common in math. Please realize these general quantities can represent millions of relationships between real quantities.

In the equation

`y = 3x + 1`,

y is a function of x , since for each value of x , there is only one value of y .

If we substitute `x = 5`, we get `y = 16` and no other value.

The values of y we get depend on the values chosen for x .

Therefore, x is the independent variable and y is the dependent variable.

The force F required to accelerate an object of mass 5 kg by an acceleration of a ms -2 is given by: `F = 5a`.

Here, F is a function of the acceleration, a.

The dependent variable is F and the independent variable is a .

Function Notation

We normally write functions as: `f(x)` and read this as "function f of x ".

We can use other letters for functions, like g ( x ) or y ( x ).

When we are solving real problems, we use meaningful letters like

P ( t ) for power at time t , F ( t ) for force at time t , h ( x ) for height of an object, x horizontal units from a fixed point.

We often come across functions like: y = 2 x 2 + 5 x + 3 in math.

We can write this using function notation:

y = f ( x ) = 2 x 2 + 5 x + 3

Function notation is all about substitution.

The value of this function f ( x ) when `x = 0` is written as `f(0)`. We calculate its value by substituting as follows:

f (0) = 2(0) 2 + 5(0) + 3 = 0 + 0 + 3 = 3

Function Notation: In General

In general, the value of any function f ( x ) when x = a is written as f ( a ).

If we have `f(x) = 4x + 10`, the value of `f(x)` for `x = 3` is written:

`f(3) = 4 × 3 + 10 = 22`

In other words, when `x = 3`, the value of the function f ( x ) is `22`.

Mathematical Notation

Mathematics is often confusing because of the way it is written.

We write `5(10)` and it means `5 × 10= 50`.

But if we write `a(10)`, this could mean, depending on the situation,

"function a of `10`" (that is, the value of the function a when the independent variable is `10`)

Or it could mean multiplication, as in:

`a × 10 = 10a`.

You have to be careful with this.

Also, be careful when substituting letters or expressions into functions.

See a discussion on this: Towards more meaningful math notation .

This example involves some fixed constant, d .

If `h(x) = dx^3+ 5x` then value of `h(x)` for `x = 10` is:

`h(10) = d(10)^3+ 5(10)` `= 1000d + 50`

We leave the d there because we don't know anything about its value.

This example involves the value of a function when the independent variable contains a constant.

If the height of an object at time t is given by

h ( t ) = 10 t 2 − 2 t , then

a. The height at time `t = 4` is

h (4) = 10(4) 2 − 2(4) = 10 ×16 − 8 = 152

b. The height at time t = b is

h ( b ) = 10 b 2 − 2 b

c. The height at time `t = 3b` is

h (3 b ) = 10(3 b ) 2 − 2(3 b ) = 10 × 9 b 2 − 6 b = 90 b 2 − 6 b

d. The height at time `t = b + 1` is

h ( b + 1) = 10( b + 1) 2 − 2( b + 1) = 10 × ( b 2 + 2 b + 1) − 2 b − 2 = 10 b 2 + 20 b + 10 − 2 b − 2 = 10 b 2 + 18 b + 8

Evaluate the following functions:

(1) Given `f(x) = 3x + 20`, find

a. `f(-4)` b. `f(10)`

a. f (−4)

= 3(−4) + 20 = −12 + 20 = 8

= 3(10) + 20 = 30 + 20 = 50

(2) Given that the height of a particular object at time t is

h ( t ) = 50 t − 4.9 t 2 , find

a. `h(2)` b. `h(5)`

= 50(2) − 4.9(2) 2 = 100 − 19.6 = 80.4

= 50(5) − 4.9(5) 2 = 250 − 122.5 = 127.5

(3) The voltage, V , in a particular circuit is a function of time t , and is given by:

V ( t ) = 3 t − 1.02 t

Find the voltage at time

a. `t = 4` b. `t = c + 10`

= 3(4) − 1.02 4 = 12 − 1.08243216 = 10.9175678

b. V ( c + 10)

= 3( c + 10) − 1.02 c + 10 = 3 c + 30 − 1.02 c + 10

(4) If F ( t ) = 3 t − t 2 for t ≤ 2 , find F (2) and F (3) .

F ( t ) = 3 t − t 2

F (2) = 3(2) − (2) 2

= 6 − 4

F (3) is not defined since `t ≤ 2`.

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An Introduction to Functions

Related Pages Inverse Functions More Lessons for Intermediate Algebra More Lessons for Algebra Math Worksheets

A series of free, online Intermediate Algebra Lessons or Algebra II lessons. Examples, solutions, videos, worksheets, and activities to help Algebra students.

In these lessons, we will learn

• how to define a function
• how to find the domain and range of a function
• how to use the Vertical Line Test
• how to define function notation
• how to calculate the composition of functions

The following diagrams show the Vertical Line Test that can be used to test whether a given graph is a function. Scroll down the page for more examples and solutions on functions and how to use the Vertical Line Test.

Introduction to Functions

In mathematics, a relationship describes one quantity in terms of another. A function is a type of relationship in which for each first component there is one and only one second component. In mathematics, an introduction to functions and how to identify whether or not a relationship is a function is very important building block since a lot of complex topics in upper-level math involve functions.

Domain and Range

An important part of understanding functions is understanding their domain and range. Domain and range are all the possible x-values and y-values of the function, and can often be described easily by looking at a graph. In order to grasp domain and range, students must understand how to determine if a relation is a function and interpreting graphs.

This video introduces the definition of a function, domain, and range.

This video shows how to graph a function and how to determine the domain and range of a function.

Vertical Line Test

Determine if a Relation is a Function

Determining Domain and Range

Function Notation

Throughout mathematics, we find function notation. Function notation is a way to write functions that is easy to read and understand. Functions have dependent and independent variables, and when we use function notation the independent variable is commonly x, and the dependent variable is F(x). In order to write a relation or equation using function notation, we first determine whether the relation is a function.

How to define function notation.

Function Notation - A basic description of function notation and a few examples involving function notation.

Composition of Functions

When we put two functions together, we have something called a composition of functions. For example, the expression g(f(x)) states that we should put the “f” function into the “g” function. To do this, we simply substitute the entire inner function into each of the variables in the outer function.

How to calculate the composition of functions.

Composition of Functions - Numerous examples are shown of how to compose functions

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Introduction to Functions

Chapter outline.

Toward the end of the twentieth century, the values of stocks of Internet and technology companies rose dramatically. As a result, the Standard and Poor’s stock market average rose as well. The graph above tracks the value of that initial investment of just under $100 over the 40 years. It shows that an investment that was worth less than $500 until about 1995 skyrocketed up to about $1100 by the beginning of 2000. That five-year period became known as the “dot-com bubble” because so many Internet startups were formed. As bubbles tend to do, though, the dot-com bubble eventually burst. Many companies grew too fast and then suddenly went out of business. The result caused the sharp decline represented on the graph beginning at the end of 2000.

Notice, as we consider this example, that there is a definite relationship between the year and stock market average. For any year we choose, we can determine the corresponding value of the stock market average. In this chapter, we will explore these kinds of relationships and their properties.

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Unit 8: Functions

About this unit.

A function is like a machine that takes an input and gives an output. Let's explore how we can graph, analyze, and create different types of functions.

Evaluating functions

• What is a function? (Opens a modal)
• Worked example: Evaluating functions from equation (Opens a modal)
• Worked example: Evaluating functions from graph (Opens a modal)
• Evaluating discrete functions (Opens a modal)
• Worked example: evaluating expressions with function notation (Opens a modal)
• Evaluate functions Get 3 of 4 questions to level up!
• Evaluate functions from their graph Get 3 of 4 questions to level up!
• Evaluate function expressions Get 3 of 4 questions to level up!

Inputs and outputs of a function

• Worked example: matching an input to a function's output (equation) (Opens a modal)
• Worked example: matching an input to a function's output (graph) (Opens a modal)
• Worked example: two inputs with the same output (graph) (Opens a modal)
• Function inputs & outputs: equation Get 3 of 4 questions to level up!
• Function inputs & outputs: graph Get 3 of 4 questions to level up!

Functions and equations

• Equations vs. functions (Opens a modal)
• Obtaining a function from an equation (Opens a modal)
• Function rules from equations Get 3 of 4 questions to level up!

Interpreting function notation

• Function notation word problem: bank (Opens a modal)
• Function notation word problem: beach (Opens a modal)
• Function notation word problems Get 3 of 4 questions to level up!

Introduction to the domain and range of a function

• Intervals and interval notation (Opens a modal)
• What is the domain of a function? (Opens a modal)
• What is the range of a function? (Opens a modal)
• Worked example: domain and range from graph (Opens a modal)
• Domain and range from graph Get 5 of 7 questions to level up!

Determining the domain of a function

• Determining whether values are in domain of function (Opens a modal)
• Examples finding the domain of functions (Opens a modal)
• Worked example: determining domain word problem (real numbers) (Opens a modal)
• Worked example: determining domain word problem (positive integers) (Opens a modal)
• Worked example: determining domain word problem (all integers) (Opens a modal)
• Identifying values in the domain Get 3 of 4 questions to level up!
• Determine the domain of functions Get 3 of 4 questions to level up!
• Function domain word problems Get 3 of 4 questions to level up!

Recognizing functions

• Recognizing functions from graph (Opens a modal)
• Does a vertical line represent a function? (Opens a modal)
• Recognizing functions from table (Opens a modal)
• Recognizing functions from verbal description (Opens a modal)
• Recognizing functions from verbal description word problem (Opens a modal)
• Recognize functions from graphs Get 3 of 4 questions to level up!
• Recognize functions from tables Get 3 of 4 questions to level up!

Maximum and minimum points

• Introduction to minimum and maximum points (Opens a modal)
• Worked example: absolute and relative extrema (Opens a modal)
• Relative maxima and minima Get 3 of 4 questions to level up!
• Absolute maxima and minima Get 3 of 4 questions to level up!

Intervals where a function is positive, negative, increasing, or decreasing

• Increasing, decreasing, positive or negative intervals (Opens a modal)
• Worked example: positive & negative intervals (Opens a modal)
• Positive and negative intervals Get 3 of 4 questions to level up!
• Increasing and decreasing intervals Get 3 of 4 questions to level up!

Interpreting features of graphs

• Graph interpretation word problem: temperature (Opens a modal)
• Graph interpretation word problem: basketball (Opens a modal)
• Creativity break: How can people get creative in algebra (Opens a modal)
• Graph interpretation word problems Get 3 of 4 questions to level up!

Average rate of change

• Introduction to average rate of change (Opens a modal)
• Worked example: average rate of change from graph (Opens a modal)
• Worked example: average rate of change from table (Opens a modal)
• Average rate of change: graphs & tables Get 3 of 4 questions to level up!

Average rate of change word problems

• Average rate of change word problem: table (Opens a modal)
• Average rate of change word problem: graph (Opens a modal)
• Average rate of change review (Opens a modal)
• Average rate of change word problems Get 3 of 4 questions to level up!

Intro to inverse functions

• Intro to inverse functions (Opens a modal)
• Inputs & outputs of inverse functions (Opens a modal)
• Graphing the inverse of a linear function (Opens a modal)
• Finding inverse functions: linear (Opens a modal)
• Functions: FAQ (Opens a modal)
• Evaluate inverse functions Get 3 of 4 questions to level up!
• Finding inverses of linear functions Get 3 of 4 questions to level up!

• Avoiding Common Math Mistakes-Expanding
• Avoiding Common Math Mistakes-Trigonometry
• Avoiding Common Math Mistakes-Simplifiying
• Avoiding Common Math Mistakes-Square Roots
• Avoiding Common Math Mistakes-Working with negatives
• Complex Numbers
• Decimal and Percent
• Dosage Calculations
• Adding and Subtracting Fractions
• BEDMAS with Fractions
• Multiplying and Dividing Fractions
• Long Division
• Long Multiplication
• Order of Operations
• Calculating Slope Examples
• Graphs of Functions
• Least Squares Trendline and Correlation
• Semi-Log and Log-Log Graphs
• Pythagorean Theorem
• Ratio and Proportion
• Rounding and Significant Figures
• Scientific Notation
• Square Root
• Unit Conversion for the Sciences
• Unit Conversion Examples
• Application of Derivatives: Examples
• Chain Rule: Examples
• Higher Order Derivatives: Examples
• Power Rule: Example
• Product Rule: Examples
• Quotient Rule: Examples
• Fundamental Theorem of Calculus
• Net Change Theorem: Example
• Newton's Method
• Completing the Square
• Simplifying Expressions
• Absolute Value Equations
• The Quadratic Formula
• Rational Equations
• Solving Equations: Application
• Solving Linear Equations
• Solving Linear Inequalities
• Solving Linear Systems
• Word Problems
• Domain and Range of Exponential and Logarithmic Functions
• Transformation of Exponential and Logarithmic Functions
• Solving Exponential and Logarithmic Equations
• Logarithmic Models
• Composition of Functions
• Domain and Range Examples
• Domain and Range Exponential and Logarithmic Fuctions
• Domain and Range of Trigonometric Functions
• Evaluating Functions
• One-to-One and Onto Functions
• Inverse Functions
• Equations of Lines
• Setting Up Linear Models
• Piecewise-Defined Functions
• Transformations of Exponential and Logarithmic Functions
• Transformations of Trigonometric Functions
• Bar Graph and Pie Chart
• Linear Regression and Correlation
• Normal Distribution
• Standard Deviation
• Avoiding Common Math Mistakes in Trigonometry
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• Trigonometry on the Unit Circle
• Introduction to Trigonometric Functions
• Inverse Trigonometric Functions
• Setting Up Trigonometric Models
• Vector Magnitude, Direction, and Components
• Angle Between Vectors
• Vector Addition, Subtraction, and Scalar Multiplication
• Vector Dot Product and Cross Product
• Matrix Addition, Subtraction, and Multiplication by a Scalar
• Matrix Multiplication
• Special Matrices and Definitions
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• Open Educational Resources
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Introduction to Functions

Let’s review some background material that you will need to study functions!

Definition of a function

Before we can begin to evaluate and work with functions, it is important to understand what a function is. Functions arise when one quantity is dependent on another. A function f is a rule that assigns to each element from one set exactly one element from another set.

There are 4 ways that a function can be represented:

• Verbally by a description in words.
• Numerically by a table of values.
• Graphically or visually using a graph.
• Algebraically by an equation.

Vertical Line Test

All relationships are not necessarily functions. Remember that we want each element in the first set to correspond to only one element in the second set. One method for testing to determine if a relationship is a function is the vertical line test. If a vertical line intersects a graph at more than one point, then the graph is not the graph of a function.

Example: Use the vertical line test to determine if the following graphs represent functions.

Solution: The figure on the left is an example of a function because no vertical line can intersect the graph at more than one point (assuming the same periodic behavior continues). The figure on the right is not an example of a function because there are many vertical lines that could be drawn that would intersect the graph at two points. 

Function Notation

In many cases, a function will be written in the form of an equation, such as f(x) = 3x + 6 where f(x) is read “f of x”. Another common way to write a function is in the form y = 3x +6. In this case, x and y are the variables, where x is the independent variable and y is the dependent variable. This simply means that making a change to x will result in a change to y. Both x and y are very common variables that are often used, but be aware that other variables may be used instead.  

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6.1: Introduction to Functions

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• Page ID 7067

• Ted Sundstrom
• Grand Valley State University via ScholarWorks @Grand Valley State University

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Exercise \(\PageIndex{1}\)

Preview Activity 1 (Functions from Previous Courses) One of the most important concepts in modern mathematics is that of a function . In previous mathematics courses, we have often thought of a function as some sort of input-output rule that assigns exactly one output to each input. So in this context, a function can be thought of as a procedure for associating with each element of some set, called the domain of the function , exactly one element of another set, called the codomain of the function . This procedure can be considered an input-output-rule. The function takes the input, which is an element of the domain, and produces an output, which is an element of the codomain. In calculus and precalculus, the inputs and outputs were almost always real numbers. So the notationf \(f(x) = x^2 sin x\) means the following:

• \(f\) is the name of the function.
• \(f(x)\) is a real number. It is the output of the function when the input is the real number \(x\). For example, \[\begin{array} {rcl} {f(\dfrac{\pi}{2})} &= & {(\dfrac{\pi}{2})^2 sin(\dfrac{\pi}{2})} \\ {} &= & {\dfrac{\pi ^2}{4} \cdot 1} \\ {} &= & {\dfrac{\pi ^2}{4}.} \end{array}\]

For this function, it is understood that the domain of the function is the set \(\mathbb{R}\) of all real numbers. In this situation, we think of the domain as the set of all possible inputs. That is, the domain is the set of all possible real numbers \(x\) for which a real number output can be determined.

This is closely related to the equation \(f = x^2 sin x\). With this equation, we frequently think of \(x\) as the input and \(y\) as the output. In fact, we sometimes write \(y = f(x)\). The key to remember is that a function must have exactly one output for each input. When we write an equation such as

\(y = \dfrac{1}{2} x^3 - 1,\)

we can use this equation to define \(y\) as a function of \(x\). This is because when we substitute a real number for \(x\) (the input), the equation produces exactly one real number for \(y\) (the output). We can give this function a name, such as \(g\), and write

\(y = g(x) = \dfrac{1}{2} x^3 - 1.\)

However, as written, an equation such as

• \(y^2 = x + 3\)

cannot be used to define \(y\) as a function of \(x\) since there are real numbers that can be substituted for \(x\) that will produce more than one possible value of \(y\). For example, if \(x = 1\), then \(y^2 = 4\), and \(y\) could be -2 or 2.

Which of the following equations can be used to define a function with \(x \in \mathbb{R}\) as the input and \(y \in \mathbb{R}\) as the output?

• \(y = x^2 - 2\)
• \(y = \dfrac{1}{2} x^3 - 1\)
• \(y = \dfrac{1}{2} x sin x\)
• \(x^2 + y^2 = 4\)
• \(y = 2x - 1\)
• \(y =dfrac{x}{x - 1}\)

Preview Activity 2 (Some Other Types of Functions)

The domain and codomain of the functions in Preview Activity \(\PageIndex{1}\) is the set \(\mathbb{R}\) of all real numbers, or some subset of \(\mathbb{R}\). In most of these cases, the way in which the function associates elements of the domain with elements of the codomain is by a rule determined by some mathematical expression. For example, when we say that \(f\) is the function such that

\(f(x) = \dfrac{x}{x - 1},\)

then the algebraic rule that determines the output of the function \(f\) when the input is \(x\) is \(\dfrac{x}{x - 1}\). In this case, we would say that the domain of \(f\) is the set of all real numbers not equal to 1 since division by zero is not defined.

However, the concept of a function is much more general than this. The domain and codomain of a function can be any set, and the way in which a function associates elements of the domain with elements of the codomain can have many different forms. The input-output rule for a function can be a formula, a graph, a table, a random process, or a verbal description. We will explore two different examples in this preview activity.

• Let \(b\) be the function that assigns to each person his or her birthday (month and day). The domain of the function \(b\) is the set of all people and the codomain of \(b\) is the set of all days in a leap year (i.e., January 1 through December 31, including February 29). (a) Explain why \(b\) really is a function. We will call this the birthday function . (b) In 1995, Andrew Wiles became famous for publishing a proof of Fermat’s Last Theorem. (See A. D. Aczel, Fermat’s Last Theorem: Unlocking the Secret of an Ancient Mathematical Problem, Dell Publishing, New York, 1996.) Andrew Wiles’s birthday is April 11, 1953. Translate this fact into functional notation using the “birthday function” \(b\). That is, fill in the spaces for the following question marks: \[b(?) = ?.\] (c) Is the following statement true or false? Explain. For each day \(D\) of the year, there exists a person \(x\) such that \(b(x) = D\). (d) Is the following statement true or false? Explain. For any people \(x\) and \(y\), if \(x\) and \(y\) are different people, then \(b(x) \ne b(y)\).
• Let \(s\) be the function that associates with each natural number the sum of its distinct natural number divisors. This is called the sum of the divisors function . For example, the natural number divisors of 6 are 1, 2, 3, and 6, and so \[\begin{array} {rcl} {s(6)} &= & {1 + 2 + 3 + 6} \\ {} &= & {12.} \end{array}\] (a) Calculate \(s(k)\) for each natural number \(k\) from 1 through 15. (b) Does there exist a natural number \(n\) such that \(s(n) = 5\)? Justify your conclusion. (c) Is it possible to find two different natural numbers \(m\) and \(n\) such that \(s(m) = s(n)\)? Explain. (d) Use your responses in (b) and (c) to determine whether the following statements true or false. i. For each \(m \in \mathbb{N}\), there exists a natural number \(n\) such that \(s(n) = m\). ii. For all \(m, n \in \mathbb{N}\), if \(m \ne n\), then \(s(m) \ne s(n)\).

The Definition of a Function

The concept of a function is much more general than the idea of a function used in calculus or precalculus. In particular, the domain and codomain do not have to be subsets of \(\mathbb{R}\). In addition, the way in which a function associates elements of the domain with elements of the codomain can have many different forms. This input-output rule can be a formula, a graph, a table, a random process, a computer algorithm, or a verbal description. Two such examples were introduced in Preview Activity \(\PageIndex{2}\).

For the birthday function , the domain would be the set of all people and the codomain would be the set of all days in a leap year. For the sum of the divisors function , the domain is the set \(\mathbb{N}\) of natural numbers, and the codomain could also be \(\mathbb{N}\). In both of these cases, the input-output rule was a verbal description of how to assign an element of the codomain to an element of the domain.

We formally define the concept of a function as follows:

A function from a set \(A\) to a set \(B\) is a rule that associates with each element \(x\) of the set \(A\) exactly one element of the set \(B\). A function from \(A\) to \(B\) is also called a mapping from \(A\) to \(B\).

Function Notation . When we work with a function, we usually give it a name. The name is often a single letter, such as \(f\) or \(g\). If \(f\) is a function from the set \(A\) to be the set \(B\), we will write \(f: A \to B\). This is simply shorthand notation for the fact that \(f\) is a function from the set \(A\) to the set \(B\). In this case, we also say that \(f\) maps \(A\) to \(B\).

Let \(f: A \to B\). (This is read, “Let \(f\) be a function from \(A\) to \(B\).”) The set \(A\) is called the domain of the function \(f\), and we write \(A = dom(f)\). The set \(B\) is called the codomain of the function \(f\), and we write \(B = codom(f)\).

If \(a \in A\), then the element of \(B\) that is associated with \(a\) is denoted by \(f(a)\) and is called the image of a under \(f\) . If \(f(a) = b\), with \(b \in B\), then a is called a preimage of \(b\) under \(f\) .

Some Function Terminology with an Example. When we have a function \(f: A \to B\), we often write \(y = f(x)\). In this case, we consider \(x\) to be an unspecified object that can be chosen from the set \(A\), and we would say that \(x\) is the independent variable of the function \(f\) and \(y\) is the dependent variable of the function \(f\).

For a specific example, consider the function \(g: \mathbb{R} \to \mathbb{R}\), where \(g(x)\) is defined by the formula

\(g(x) = x^2 - 2.\)

Note that this is indeed a function since given any input \(x\) in the domain, \(\mathbb{R}\), there is exactly one output \(g(x)\) in the codomain, \(\mathbb{R}\). For example,

\[\begin{array} {rcl} {g(-2)} &= & {(-2)^2 - 2 = 2,} \\ {g(5)} &= & {5^2 - 2 = 23,} \\ {g(\sqrt 2)} &= & {(\sqrt 2)^2 - 2 = 0,} \\ {g(-\sqrt 2)} &= & {(-\sqrt 2)^2 - 2 = 0.} \end{array}\]

So we say that the image of -2 under \(g\) is 2, the image of 5 under \(g\) is 23, and so on.

Notice in this case that the number 0 in the codomain has two preimages, \(-\sqrt 2\) and \(\sqrt 2\). This does not violate the mathematical definition of a function since the definition only states that each input must produce one and only one output. That is, each element of the domain has exactly one image in the codomain. Nowhere does the definition stipulate that two different inputs must produce different outputs.

Finding the preimages of an element in the codomain can sometimes be difficult. In general, if \(y\) is in the codomain, to find its preimages, we need to ask, “For which values of \(x\) in the domain will we have \(y = g(x)\)?” For example, for the function g, to find the preimages of 5, we need to find all \(x\) for which \(g(x) = 5\). In this case, since \(g(x) = x^2 - 2\), we can do this by solving the equation

\(x^2 - 2 = 5.\)

The solutions of this equation are \(-\sqrt 7\) and \(\sqrt 7\). So for the function \(g\), the preimages of 5 are \(-\sqrt 7\) and \(\sqrt 7\). We often use set notation for this and say that the set of preimages of 5 for the function \(g\) is {\(-\sqrt 7\), \(\sqrt 7\)}.

Also notice that for this function, not every element in the codomain has a preimage. For example, there is no input \(x\) such that \(g(x) = -3\). This is true since for all real numbers \(x\), \(x^2 \ge 0\) and hence \(x^2 - 2 \ge -2\). This means that for all \(x\) in \(\mathbb{R}\), \(g(x) \ge -2\).

Finally, note that we introduced the function g with the sentence, “Consider the function \(g: \mathbb{R} \to \mathbb{R}\), where \(g(x)\) is defined by the formula \(g(x) = x^2 - 2\).” This is one correct way to do this, but we will frequently shorten this to, “Let \(g: \mathbb{R} \to \mathbb{R}\) be defined by \(g(x) = x^2 - 2\)”, or “Let \(g: \mathbb{R} \to \mathbb{R}\), where \(g(x) = x^2 - 2\).”

Progress Check 6.1 (Images and Preimages)

Let \(f: \mathbb{R} \to \mathbb{R}\) be defined by \(f(x) = x^2 - 5x\) for all \(x \in \mathbb{R}\). and let \(g: \mathbb{Z} \to \mathbb{Z}\) be defined by \(g(m) = m^2 - 5m\) for all \(m \in \mathbb{Z}\).

• Determine \(f(-3)\) and \(f(\sqrt 8)\).
• Determine \(g(2)\) and \(g(-2)\).
• Determine the set of all preimage of 6 for the function \(f\).
• Determine the set of all preimage of 6 for the function \(g\).
• Determine the set of all preimage of 2 for the function \(f\).
• Determine the set of all preimage of 2 for the function \(g\).

Add texts here. Do not delete this text first.

The Codomain and Range of a Function

Besides the domain and codomain, there is another important set associated with a function. The need for this was illustrated in the example of the function \(g\) on page 285. For this function, it was noticed that there are elements in the codomain that have no preimage or, equivalently, there are elements in the codomain that are not the image of any element in the domain. The set we are talking about is the subset of the codomain consisting of all images of the elements of the domain of the function, and it is called the range of the function.

Let \(f: A \to B\). The set \(\{f(x)\ |\ x \in A\}\) is called the range of the function \(f\) and is denoted by range (\(f\)). The range of \(f\) is sometimes called the image of the function \(f\) (or the image of \(A\) under \(f\) ).

The range of \(f: A \to B\) could equivalently be defined as follows:

range(\(f\))\( = \{y \in B\ |\ y = f(x) \text{ for some } x \in A\}\).

Notice that this means that range(\(f\)) \(\subseteq\) codom(\(f\)) but does not necessarily mean that range(\(f\)) \(=\) codom(\(f\)). Whether we have this set equality or not depends on the function \(f\). More about this will be explored in Section 6.3.

Progress Check 6.2 (Codomain and Range)

• Let \(b\) be the function that assigns to each person his or her birthday (month and day). (a) What is the domain of this function? (b) What is a codomain for this function? (c) In Preview Activity \(\PageIndex{2}\), we determined that the following statement is true: For each day \(D\) of the year, there exists a person \(x\) such that \(b(x) = D\). What does this tell us about the range of the function \(b\)? Explain.
• Let \(s\) be the function that associates with each natural number the sum of its distinct natural number factors. (a) What is the domain of this function? (b) What is a codomain for this function? (c) In Preview Activity \(\PageIndex{2}\), we determined that the following statement is false: For each \(m \in \mathbb{N}\), there exists a natural number \(n\) such that \(s(n) = m\). Give an example of a natural number \(m\) that shows this statement is false, and explain what this tells us about the range of the function \(s\).

The Graph of a Real Function

We will finish this section with methods to visually communicate information about two specific types of functions. The first is the familiar method of graphing functions that was a major part of some previous mathematics courses. For example, consider the function \(g: \mathbb{R} \to \mathbb{R}\) defined by \(g(x) = x^2 - 2x - 1\).

Every point on this graph corresponds to an ordered pair (\(x\), \(y\)) of real numbers, where \(y = g(x) = x^2 - 2x - 1\). Because we use the Cartesian plane when drawing this type of graph, we can only use this type of graph when both the domain and the codomain of the function are subsets of the real numbers \(\mathbb{R}\). Such a function is sometimes called a real function . The graph of a real function is a visual way to communicate information about the function. For example, the range of \(g\) is the set of all y-values that correspond to points on the graph. In this case, the graph of \(g\) is a parabola and has a vertex at the point (1, -2). ( Note : The x-coordinate of the vertex can be found by using calculus and solving the equation \(f\prime (x) = 0\).) Since the graph of the function \(g\) is a parabola, we know that pattern shown on the left end and the right end of the graph continues and we can conclude that the range of \(g\) is the set of all \(y \in \mathbb{R}\) such that \(y \ge -2\). That is,

range(\(g\))\( = \{y \in \mathbb{R}\ |\ y \ge -2\}.\)

Progress Check 6.3 (Using the Graph of a Real Function)

The graph in Figure 6.2 shows the graph of (slightly more than) two complete periods for a function \(f: \mathbb{R} \to \mathbb{R}\), where \(f(x) = Asin(Bx)\) for some positive real number constants \(A\) and \(B\).

• We can use the graph to estimate the output for various inputs. This is done by estimating the \(y\)-coordinate for the point on the graph with a specified \(x\)-coordinate. On the graph, draw vertical lines at \(x = -1\) and \(x = 2\) and estimate the values of \(f(-1)\) and \(f(2)\).
• Similarly, we can estimate inputs of the function that produce a specified output. This is done by estimating the \(x\)-coordinates of the points on the graph that have a specified \(y\)-coordinate. Draw a horizontal line at \(y = 2\) and estimate at least two values of \(x\) such that \(f(x) = 2\).
• Use the graph Figure 6.2 to estimate the range of the function \(f\).

Arrow Diagrams

Sometimes the domain and codomain of a function are small, finite sets. When this is the case, we can define a function simply by specifying the outputs for each input in the domain. For example, if we let \(A = \{1, 2, 3\}\) and let \(B = \{a, b\}\), we can define a function \(F: A \to B\) by specifying that

\(F(1) = a, F(2) = a,\text{ and } F(3) = b.\)

This is a function since each element of the domain is mapped to exactly one element in \(B\). A convenient way to illustrate or visualize this type of function is with a so-called arrow diagram as shown in Figure 6.3. An arrow diagram can

be used when the domain and codomain of the function are finite (and small). We represent the elements of each set with points and then use arrows to show how the elements of the domain are associated with elements of the codomain. For example, the arrow from the point 2 in \(A\) to the point \(a\) in \(B\) represents the fact that \(F(2) = a\). In this case, we can use the arrow diagram in Figure 6.3 to conclude that range(\(F\))\( = \{a, b\}\).

Progress Check 6.4 (Working with Arrow Diagrams)

Let \(A = \{1, 2, 3, 4\}\) and let \(B = \{a, b, c\}\).

• Which of the arrow diagrams in Figure 6.4 can be used to represent a function from \(A\) to \(B\)? Explain.

Exercises 6.1

• Let \(f: \mathbb{R} \to \mathbb{R}\) be defined by \(f(x) = x^2 - 2x\). (a) Evaluate \(f(-3)\), \(f(-1)\), \(f(1)\), and \(f(3)\). (b) Determine the set of all of the preimages of 0 and the set of all of the preimages of 4. (c) Sketch a graph of the function \(f\). (d) Determine the range of the function \(f\).
• Let \(\mathbb{R} ^{\ast} = \{x \in \mathbb{R}\ |\ x \ge 0\}\), and let \(s: \mathbb{R} \to \mathbb{R} ^{\ast}\) be defined by \(s(x) = x^2\). (a) Evaluate \(s(-3)\), \(s(-1)\), \(s(1)\), and \(s(3)\). (b) Determine the set of all of the preimages of 0 and the set of all of the preimages of 2. (c) Sketch a graph of the function \(s\). (d) Determine the range of the function \(s\).
• Let \(f: \mathbb{Z} \to \mathbb{Z}\) be defined by \(f(m) = 3 - m\). (a) Evaluate \(f(-7)\), \(f(-3)\), \(f(3)\), and \(f(7)\). (b) Determine the set of all of the preimages of 5 and the set of all of the preimages of 4. (c) Determine the range of the function \(f\). (d) This function can be considered a real function since \(\mathbb{Z} \subseteq \mathbb{R}\). Sketch a graph of this function. Note : The graph will be an infinite set of points that lie on a line. However, it will not be a line since its domain is not \(\mathbb{R}\) but is \(\mathbb{Z}\).
• Let \(f: \mathbb{Z} \to \mathbb{Z}\) be defined by \(f(m) = 2m + 1\). (a) Evaluate \(f(-7)\), \(f(-3)\), \(f(3)\), and \(f(7)\). (b) Determine the set of all of the preimages of 5 and the set of all of the preimages of 4. (c) Determine the range of the function \(f\). (d) Sketch a graph of the function \(f\). See the comments in Exercise (3d).
• Recall that a real function is a function whose domain and codomain are subsets of the real numbers R. (See page 288.) Most of the functions used in calculus are real functions. Quite often, a real function is given by a formula or a graph with no specific reference to the domain or the codomain. In these cases, the usual convention is to assume that the domain of the real function \(f\) is the set of all real numbers \(x\) for which \(f(x)\) is a real number, and that the codomain is \(\mathbb{R}\). For example, if we define the (real) function \(f\) by \[f(x) = \dfrac{x}{x - 2},\] we would be assuming that the domain is the set of all real numbers that are not equal to 2 and that the codomain in \(\mathbb{R}\). Determine the domain and range of each of the following real functions. It might help to use a graphing calculator to plot a graph of the function. (a) The function \(k\) defined by \(k(x) = \sqrt{x - 3}\) (b) The function \(F\) defined by \(F(x) = ln(2x - 1)\) (c) The function \(f\) defined by \(f(x) = 3sin(2x)\) (d) The function \(g\) defined by \(g(x) = \dfrac{4}{x^2 - 4}\) (e) The function \(G\) defined by \(G(x) = 4cos(\pi x) + 8\)
• The number of divisors function . Let \(d\) be the function that associates with each natural number the number of its natural number divisors. That is \(d: \mathbb{N} \to \mathbb{N}\) where \(d(n)\) is the number of natural number divisors of \(n\). For example, \(d(6) = 4\) since 1, 2, 3, and 6 are the natural number divisors of 6. (a) Calculate \(d(k)\) for each natural number \(k\) from 1 through 12. (b) Does there exist a natural number \(n\) such that \(d(n) = 1\)? What is the set of preimages of the natural number 1. (c) Does there exist a natural number \(n\) such that \(d(n) = 2\)? If so, determine the set of all preimages of the natural number 2. (d) Is the following statement true or false? Justify your conclusion. For all \(m, n \in \mathbb{N}\), if \(m \ne n\), then \(d(m) \ne d(n)\). (e) Calculate \(d(2^k)\) for \(k = 0\) and for each natural number \(k\) from 1 through 6. (f) Based on your work in Exercise (6e), make a conjecture for a formula for \(d(2^n)\) where \(n\) is a nonnegative integer. Then explain why your conjecture is correct. (g) Is the following statement is true or false? For each \(n \in \mathbb{N}\), there exists a natural number \(m\) such that \(d(m) = n\).
• In Exercise (6), we introduced the number of divisors function \(d\). For this function, \(d: \mathbb{N} \to \mathbb{N}\), where \(d(n)\) is the number of natural number divisors of \(n\). A function that is related to this function is the so-called set of divisors function . This can be defined as a function \(S\) that associates with each natural number the set of its distinct natural number factors. For example, \(S(6) = \{1, 2, 3, 6\}\) and \(S(10) = \{1, 2, 5, 10\}\). (a) Discuss the function \(S\) by carefully stating its domain, codomain, and its rule for determining outputs. (b) Determine \(S(n)\) for at least five different values of \(n\). (c) Determine \(S(n)\) for at least three different prime number values of \(n\). (d) Does there exist a natural number \(n\) such that card(\(S(n) = 1\))? Explain. [Recall that card(\(S(n)\)) is the number of elements in the set \(S(n)\).] (e) Does there exist a natural number \(n\) such that card(\(S(n) = 2\))? Explain. (f) Write the output for the function \(d\) in terms of the output for the function \(S\). That is, write \(d(n)\) in terms of \(S(n)\). (g) Is the following statement true or false? Justify your conclusion. For all natural numbers \(m\) and \(n\), if \(m \ne n\), then \(S(m) \ne S(n)\). (h) Is the following statement true or false? Justify your conclusion. For all sets \(T\) that are subsets of \(\mathbb{N}\), there exists a natural number \(n\) such that \(S(n) = T\). Explorations and Activities
• Creating Functions with Finite Domains. Let \(A = \{a, b, c, d\}\), \(B= \{a, b, c\}\). and \(C = \{s, t, u, v\}\). In each of the following exercises, draw an arrow diagram to represent your function when it is appropriate. (a) Create a function \(f: A \to C\) whose range is the set \(C\) or explain why it is not possible to construct such a function. (b) Create a function \(f: A \to C\) whose range is the set \(\{u,v\}\) or explain why it is not possible to construct such a function. (c) Create a function \(f: B \to C\) whose range is the set \(C\) or explain why it is not possible to construct such a function. (d) Create a function \(f: A \to C\) whose range is the set \(\{u\}\) or explain why it is not possible to construct such a function. (e) If possible, create a function \(f: A \to C\) that satisfies the following condition: For all \(x,y \in A\), if \(x \ne y\), then \(f(x) \ne f(y)\). If it is not possible to create such a function, explain why. (f) If possible, create a function \(f: A \to \{s, t, u\}\) that satisfies the following condition: For all \(x,y \in A\), if \(x \ne y\), then \(f(x) \ne f(y)\). If it is not possible to create such a function, explain why.