lesson 2 homework practice function rules

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

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

Using Function Notation to Describe Rules (Part 2)

Lesson Narrative

In an earlier lesson, students learned that some functions can be defined with a rule and the rule can be expressed using function notation. In this lesson, students use rules of functions to find the output when the input is given (or to evaluate functions) and to find the input when the output is known (or to solve equations that define functions). They also interpret rules of functions in terms of a situation. Along the way, they practice reasoning quantitatively and abstractly (MP2).

The term linear function is introduced here. In middle school, students learned a relationship between two quantities is linear if one quantity changes at a constant rate relative to the other. Students see that a linear function can be understood in similar terms: a function is linear if the output changes by a constant rate relative to its input.

This lesson includes an optional activity that is designed to enable students to use technology to graph and evaluate functions expressed in function notation. This skill can help to develop students’ understanding of functions and ability to solve problems in this unit and in future units.

Learning Goals

Teacher Facing

• Evaluate functions and solve equations given in function notation, either by graphing or by reasoning algebraically.
• Understand a linear function as a function whose output changes at a constant rate and whose graph is a line.
• Use technology to graph and evaluate functions given in function notation.

Student Facing

• Let’s graph and find the values of some functions.

Required Materials

• Graphing technology

Required Preparation

For the optional activity Function Notation and Graphing Technology, acquire devices that can run Desmos (recommended) or other graphing technology. It is ideal if each student has their own device. (Desmos is available under Math Tools.)

Learning Targets

• I can use technology to graph a function given in function notation, and use the graph to find the values of the function.
• I know different ways to find the value of a function and to solve equations written in function notation.
• I know what makes a function a linear function.

CCSS Standards

• HSA-REI.A.1

Building Towards

Glossary Entries

A linear function is a function that has a constant rate of change. Another way to say this is that it grows by equal differences over equal intervals. For example,  \(f(x)=4x-3\)  defines a linear function. Any time \(x\) increases by 1, \(f(x)\) increases by 4.

Print Formatted Materials

Teachers with a valid work email address can  click here to register or sign in for free access to Cool Down, Teacher Guide, and PowerPoint materials.

Additional Resources

Math Worksheets Land

Math Worksheets For All Ages

• Math Topics
• Grade Levels

Solving Function Tables Worksheets

A function is a rule that designates a fixed input to a fixed output. For every different input there is there is a unique corresponding output. We can represent the various inputs and outputs of functions in the form of a function table. This is just one of several different methods that we can use to describe a function, but it is highly effective for communicating them to other people. These worksheets and lessons show students their way around function tables and their uses. The have a great number of applications that can help you describe processes quicker and easier.

Aligned Standard: 8.F.A.1

• Tables Step-by-Step Lesson - We give you a rule set and an x,y table. Go crazy!
• Guided Lesson - More rules and more tables. This thing just got out of control!
• Guided Lesson Explanation - This is a plug and chug activity as they called it in my linear algebra class.
• Independent Practice - This sheet will help you perfect the skill to a "T", or is it tee?
• Matching Worksheet - Match the rule sets and tables to their completed versions.
• Answer Keys - These are for all the unlocked materials above.

Homework Sheets

Some tables are completely empty on the y-value, some have a few entries.

• Homework 1 - We have to complete the table to show how to fill the y block. We will use the equation y = 4x to complete the table.
• Homework 2 - Make sure to get rid of all operations first and then proceed.
• Homework 3 - Just a few missing values are in this function table.

Practice Worksheets

I always like to try to give students one complete problem. It helps those that struggle to grow.

• Practice 1 - Complete the function table for each equation. Input the x value into each equation to determine the end y value.
• Practice 2 - In each case, you will need to solve for y, but it is just a plug and chug equation.
• Practice 3 - Reverse engineer the given value to make sure the function is properly functioning.

Math Skill Quizzes

The quizzes follow the same scheme that can be seen with the homework and practice sheets.

• Quiz 1 - Some are missing three values and other are missing four. If you forgot how these work, you can use the completed problems to help you better understand it.
• Quiz 2 - Completely blank y values. There is no help given with this quiz.
• Quiz 3 - As normal, we save quiz 3 for the negative values. There is one sample value worked out for you.

How to Solve Function Tables

A function table represents the inputs and their respective outputs based on a given function that the input passes through. Function tables can be represented in both vertical and horizontal orientations in depends on which best suits your situation. In this lesson, we will take a horizontal approach. So, in the examples that we have, the functions will be displayed in two rows. One row will display the input and the other will be displaying the output functions.

Let's take example of selling or buying candy bars. We can represent that function in the form of a table. If candy bars cost $2. We can represent this in a simple equation like this: c = 2x. In this equation c would indicate the table cost and x would be the number of candy bars you could buy or sell. We can then present this in a function table to give us a series of prices that we can refer to when selling the candy.

This could be helpful for you to instantly know the price when a customer buys different numbers of candy bars. You would just take a quick glance at this table, and you would know how much to charge them. The c value indicates the number of dollars ($). We recognize that if we only have $4to spend, we can only buy 2 candy bars.

Function Table Rules - Each function is a rule, so every single function table has a certain rule that defines the relationship that exists between the output and the input. Sometimes a rule is best described in terms of the words and in some cases the best way to define it is by using an equation.

How Does This Skill Apply to Everyday Life?

Over the course of the average day you come in contact with machines that are constantly processing values through a function table and you probably do not even realize it. If you have used a vending machines to buy a bag of chips, a bottle of water, or to make your favorite coffee this type of math was involved for the machine to understand that you input enough money to make a valid transaction. You will continue to put coins or bills into the vending machine until you satisfy the rule to make the purchase. If you have ever calculated your weekly salary for a job, you have used this form of math. It is a direct function between your hourly pay rate and the number of hours that you performed the job. If you own a mobile phone that does not have an unlimited plan, the pricing is based on this form of math as well. This a more complex rule because there is a flat monthly charge and then a charge associated with number of calls or text messages that you make over the course of a month. If you think about it the number of different situations that this can be directly related to is boundless.

Get Access to Answers, Tests, and Worksheets

Become a paid member and get:

• Answer keys to everything
• Unlimited access - All Grades
• 64,000 printable Common Core worksheets, quizzes, and tests
• Used by 1000s of teachers!

Worksheets By Email:

Get Our Free Email Now!

We send out a monthly email of all our new free worksheets. Just tell us your email above. We hate spam! We will never sell or rent your email.

Thanks and Don't Forget To Tell Your Friends!

I would appreciate everyone letting me know if you find any errors. I'm getting a little older these days and my eyes are going. Please contact me, to let me know. I'll fix it ASAP.

• Privacy Policy
• Other Education Resource

© MathWorksheetsLand.com, All Rights Reserved

• Texas Go Math
• Big Ideas Math
• Engageny Math
• McGraw Hill My Math
• enVision Math
• 180 Days of Math
• Math in Focus Answer Key
• Math Expressions Answer Key
• Privacy Policy

Eureka Math Grade 8 Module 5 Lesson 2 Answer Key

Engage ny eureka math 8th grade module 5 lesson 2 answer key, eureka math grade 8 module 5 lesson 2 example answer key.

Exercises 1–5

b. Does the function D = 16t 2 accurately represent the distance the stone fell after a given time t? In other words, does the function described by this rule assign to t the correct distance? Explain. Answer: Yes, the function accurately represents the distance the stone fell after the given time interval. Each computation using the function resulted in the correct distance. Therefore, the function assigns to t the correct distance.

Exercise 5. It takes Josephine 34 minutes to complete her homework assignment of 10 problems. If we assume that she works at a constant rate, we can describe the situation using a function. a. Predict how many problems Josephine can complete in 25 minutes. Answer: Answers will vary.

b. Write the two-variable linear equation that represents Josephine’s constant rate of work. Answer: Let y be the number of problems she can complete in x minutes. \(\frac{10}{34}\) = \(\frac{y}{x}\) y = \(\frac{10}{34}\) x y = \(\frac{5}{17}\) x

d. Compare your prediction from part (a) to the number you found in the table above. Answer: Answers will vary.

e. Use the formula from part (b) to compute the number of problems completed when x = -7. Does your answer make sense? Explain. Answer: y = \(\frac{5}{17}\) (-7) = -2.06 No, the answer does not make sense in terms of the situation. The answer means that Josephine can complete -2.06 problems in -7 minutes. This obviously does not make sense.

f. For this problem, we assumed that Josephine worked at a constant rate. Do you think that is a reasonable assumption for this situation? Explain. Answer: It does not seem reasonable to assume constant rate for this situation. Just because Josephine was able to complete 10 problems in 34 minutes does not necessarily mean she spent the exact same amount of time on each problem. For example, it may have taken her 20 minutes to do 1 problem and then 14 minutes total to finish the remaining 9 problems.

Eureka Math Grade 8 Module 5 Lesson 2 Problem Set Answer Key

b. Can you determine the value of the output for an input of x = -4? If so, what is it? Answer: When x = -4, the output is -5.

c. Does an input of -4 make sense in this situation? Explain. Answer: No, an input of -4 does not make sense for the situation. It would mean -4 bags of candy. You cannot purchase -4 bags of candy.

Question 8. Each and every day a local grocery store sells 2 pounds of bananas for $1.00. Can the cost of 2 pounds of bananas be represented as a function of the day of the week? Explain. Answer: Yes, this situation can be represented by a function. Assign to each day of the week the value $1.00.

Eureka Math Grade 8 Module 5 Lesson 2 Exit Ticket Answer Key

Question 2. Kelly can tune 4 cars in 3 hours. If we assume he works at a constant rate, we can describe the situation using a function. a. Write the function that represents Kelly’s constant rate of work. Answer: Let y represent the number of cars Kelly can tune up in x hours; then \(\frac{y}{x}\) = \(\frac{4}{3}\) y = \(\frac{4}{3}\) x

c. Kelly works 8 hours per day. According to this work, how many cars will he finish tuning at the end of a shift? Answer: Using the function, Kelly will tune up 10.67 cars at the end of his shift. That means he will finish tuning up 10 cars and begin tuning up the 11th car.

d. For this problem, we assumed that Kelly worked at a constant rate. Do you think that is a reasonable assumption for this situation? Explain. Answer: No, it does not seem reasonable to assume a constant rate for this situation. Just because Kelly tuned up 4 cars in 3 hours does not mean he spent the exact same amount of time on each car. One car could have taken 1 hour, while the other three could have taken 2 hours total.

Leave a Comment Cancel Reply

You must be logged in to post a comment.

• Notifications 0

• Add Friend ($5)

As a registered member you can:

• View all solutions for free
• Request more in-depth explanations for free
• Ask our tutors any math-related question for free
• Email your homework to your parent or tutor for free
• Grade 6 McGraw Hill Glencoe - Answer Keys

Yes, email page to my online tutor. ( if you didn't add a tutor yet, you can add one here )

Thank you for doing your homework!

Submit Your Question