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College Algebra with Corequisite Support

3.1 Functions and Function Notation

College Algebra with Corequisite Support3.1 Functions and Function Notation

Learning Objectives

In this section, you will:

  • 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.

Corequisite Skills

Learning Objectives

  • Find the value of a function (IA 3.5.3)

Objective 1: Find the value of a function (IA 3.5.3)

A relation is any set of ordered pairs, (x,y). The collection of x-values in the ordered pairs together make up the domain. The collection of y-values in the ordered pairs together make up the range.

A special type of relation, called a function, is studied extensively in mathematics. A function is a relation that assigns to each element in its domain exactly one element in the range. For each ordered pair in the relation, each x-value is matched with only one y-value.

Function Notation

For the function y=f(x)y=f(x)

fis the name of the function xis the domain value f(x)is the range valueycorresponding to the valuexfis the name of the function xis the domain value f(x)is the range valueycorresponding to the valuex

We read f(x)f(x) as f of x or the value of f at x.

Representation of Functions, y = f ( x ) y = f ( x )

There are many ways to represent functions including:

  • Equations
  • Tables of input and output values
  • Collections of ordered pairs, or points (x,y) = (independent variable, dependent variable)
  • Graphs
  • Mappings
  • Verbal descriptions

Example 1

Find the value of a function

  1. For the function f(x)=2x-5f(x)=2x-5
    Find f(4)f(4) , f(-6)f(-6) , f(0)f(0) , and f(a)f(a) .
    Find the value of xx that makes f(x)=11f(x)=11
  2. Refer to the following table of values for the function g(x)g(x) .
    xx g(x)g(x)
    00 22
    11 55
    22 1414
    33 2929
    Find g(1)g(1) , g(3)g(3) , g(0)g(0) .
  3. For a man of height 511511 the mapping below shows the corresponding Body Mass Index (BMI). The body mass index is a measurement of body fat based on height and weight. A BMI of 18.524.918.524.9 is considered healthy.

    This figure shows two table that each have one column. The table on the left has the header “Weight (lbs)” and lists the numbers 130, 140, 150, 160, 170, 180, 190, and 200. The table on the right has the header “BMI” and lists the numbers 22. 3, 19. 5, 20. 9, 27. 9, 25. 1, 26. 5, 23. 7, and 18. 1. There are arrows starting at numbers in the weight table and pointing towards numbers in the BMI table. The first arrow goes from 130 to 18. 1. The second arrow goes from 140 to 19. 5. The third arrow goes from 150 to 20. 9. The fourth arrow goes from 160 to 22. 3. The fifth arrow goes from 170 to 23. 7. The sixth arrow goes from 180 to 25. 1. The seventh arrow goes from 190 to 26. 5. The eighth arrow goes from 200 to 27. 9.


    Find the BMI for a man of height 511511 who weighs 180180 pounds. Find the weight of a man of height 511511 and who has a BMI of 22.322.3 .

Practice Makes Perfect

Find the value of a function

1.
.


  1. Find: f(0)f(0) .
  2. Find the values for xx when f(x)=0f(x)=0 .
2.

For the function f(x)=3x2-2x+1f(x)=3x2-2x+1 find

  1. f(3)f(3)
  2. f(-2)f(-2)
  3. f(t)f(t)
  4. The value(s) of x that make f(x)=1f(x)=1 .
3.

For the function g(x)=-4+3x+19g(x)=-4+3x+19 , find the following. Make sure to give exact values.

  1. g(-5)g(-5)
  2. g(2)g(2)
  3. g(0)g(0)
  4. The value(s) of xx that make g(x)=0g(x)=0 .
4.
.


Use the mapping y=G(x)y=G(x) to find the following:

  1. G(0)=G(0)=
  2. G(9)=G(9)=
  3. If G(x)=9G(x)=9 , then x=x= ________
  4. Is G(x)G(x) a function? Explain why or why not.
5.
.


Use the mapping y=F(x)y=F(x) to find the following:

  1. F(-1)=F(-1)=
  2. F(4)=F(4)=
  3. F(2)=F(2)=
  4. If F(x)=9F(x)=9 , then x=x= ________
  5. Is F(x)F(x) a function? Explain why or why not.
6.

If h(x)=5x-7h(x)=5x-7
Find:

  1. h(-3)=h(-3)=
  2. h(0)=h(0)=
  3. h(w+4)=h(w+4)=
  4. h(x)=23h(x)=23 , then x=x= ________
  5. h(x)=-15h(x)=-15 , then x=x= ________
7.

g(t)=2|t-5|+4g(t)=2|t-5|+4
Find:

  1. g(-3)=g(-3)=
  2. g(0)=g(0)=
  3. g(5)=g(5)=
  4. g(w)=g(w)=
  5. h(x)=20h(x)=20 , then x=x= ________
8.
  1. Use the following description to build a function called f(x)f(x) . "An input value is squared, multiplied by -2-2 and added to 33 .”
    f(x)=f(x)=
  2. Create a table of input/output values for f(x)f(x) below. Show three numerical input values and one variable input value, and the corresponding output values in your table.
    xx f(x)f(x)
    ________ ________
    ________ ________
    ________ ________
    ________ ________

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)} {(1,2),(2,4),(3,6),(4,8),(5,10)}

The domain is {1,2,3,4,5}. {1,2,3,4,5}. The range is {2,4,6,8,10}. {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.x. Each value in the range is also known as an output value, or dependent variable, and is often labeled lowercase letter y.y.

A function f 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}, {1,2,3,4,5}, is paired with exactly one element in the range, {2,4,6,8,10}. {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

{ (odd,1),(even,2),(odd,3),(even,4),(odd,5) } { (odd,1),(even,2),(odd,3),(even,4),(odd,5) }

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

Figure 1 compares relations that are functions and not functions.

Three relations that demonstrate what constitute a function.
Figure 1 (a) This relationship is a function because each input is associated with a single output. Note that input qq and rr both give output n. n. (b) This relationship is also a function. In this case, each input is associated with a single output. (c) This relationship is not a function because input q q is associated with two different outputs.

Function

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.

  1. Identify the input values.
  2. Identify the output values.
  3. 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 1

Determining If Menu Price Lists Are Functions

The coffee shop menu, shown below, consists of items and their prices.
Is price a function of the item?
Is the item a function of the price?

A menu of donut prices from a coffee shop where a plain donut is $1.49 and a jelly donut and chocolate donut are $1.99.

Example 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 1 shows a possible rule for assigning grade points.

Percent grade 0–56 57–61 62–66 67–71 72–77 78–86 87–91 92–100
Grade point average 0.0 1.0 1.5 2.0 2.5 3.0 3.5 4.0
Table 1

Try It #1

Table 21 lists the five greatest baseball players of all time in order of rank.

Player Rank
Babe Ruth 1
Willie Mays 2
Ty Cobb 3
Walter Johnson 4
Hank Aaron 5
Table 2
  1. Is the rank a function of the player name?
  2. Is the player name a function of the 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 h for height and a a for age. The letters f,g, f,g, and h h are often used to represent functions just as we use x,y, x,y, and z z to represent numbers and A,B, A,B, and C C to represent sets.

his fof a We name the function f;height is a function of age. h=f(a) We use parentheses to indicate the function input f(a) We name the function f;the expression is read as “fof a.” his fof a We name the function f;height is a function of age. h=f(a) We use parentheses to indicate the function input f(a) We name the function f;the expression is read as “fof a.”

Remember, we can use any letter to name the function; the notation h( a )h( a ) shows us that hh depends on a. a. The value aa must be put into the function hh 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 )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 )y=f( x ) defines a function named f. f. This is read as yy is a function of x. x. The letter x x represents the input value, or independent variable. The letter y,  y,  or f( x ),f( x ), represents the output value, or dependent variable.

Example 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. Assume that the domain does not include leap years.

Analysis

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 4

Interpreting Function Notation

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

Try It #2

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

Q&A

Instead of a notation such as y=f(x), y=f(x), could we use the same symbol for the output as for the function, such as y=y(x), 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, f, which is a rule or procedure, and the output yy we get by applying ff to a particular input x. x. This is why we usually use notation such as y=f( x ),P=W( d ), 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 3 lists the input number of each month (January = 1, 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 ff where D=f( m )D=f( m ) identifies months by an integer rather than by name.

Month number, m m (input) 1 2 3 4 5 6 7 8 9 10 11 12
Days in month, D D (output) 31 28 31 30 31 30 31 31 30 31 30 31
Table 3

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

n n 1 2 3 4 5
Q Q 8 6 7 6 8
Table 4

Table 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.

Age in years, a a (input) 5 5 6 7 8 9 10
Height in inches, h h (output) 40 42 44 47 50 52 54
Table 5

How To

Given a table of input and output values, determine whether the table represents a function.

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

Example 5

Identifying Tables that Represent Functions

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

Input Output
2 1
5 3
8 6
Table 6
Input Output
–3 5
0 1
4 5
Table 7
Input Output
1 0
5 2
5 4
Table 8

Try It #3

Does Table 9 represent a function?

Input Output
1 10
2 100
3 1000
Table 9

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 )=53 x 2 f( x )=53 x 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.

  1. Substitute the input variable in the formula with the value provided.
  2. Calculate the result.

Example 6

Evaluating Functions at Specific Values

Evaluate f( x )= x 2 +3x4f( x )= x 2 +3x4 at:

  1. 2 2
  2. a a
  3. a+h a+h
  4. Now evaluate f( a+h )f( a ) h f( a+h )f( a ) h

Example 7

Evaluating Functions

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

Try It #4

Given the function g( m )= m4 ,g( m )= m4 , evaluate g( 5 ).g( 5 ).

Example 8

Solving Functions

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

Try It #5

Given the function g( m )= m4 ,g( m )= m4 , solve g( m )=2.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=122n+6p=12 expresses a functional relationship between nn and p. p. We can rewrite it to decide if pp is a function of n. n.

How To

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

  1. 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.
  2. 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 9

Finding an Equation of a Function

Express the relationship 2n+6p=122n+6p=12 as a function p=f( n ),p=f( n ), if possible.

Analysis

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

Example 10

Expressing the Equation of a Circle as a Function

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

Try It #6

If x8 y 3 =0,x8 y 3 =0, express yy as a function of x. x.

Q&A

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 ,x=y+ 2 y , if we want to express yy as a function of x,x, there is no simple algebraic formula involving only xx that equals y.y. However, each xx does determine a unique value for y,y, and there are mathematical procedures by which yy can be found to any desired accuracy. In this case, we say that the equation gives an implicit (implied) rule for yy as a function of x,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. See Table 10.2

Pet Memory span in hours
Puppy 0.008
Adult dog 0.083
Cat 16
Goldfish 2160
Beta fish 3600
Table 10

At times, evaluating a function in table form may be more useful than using equations.Here let us call the function P. 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 PP at the input value of “goldfish.” We would write P(goldfish)=2160. 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 PP 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 11

Evaluating and Solving a Tabular Function

Using Table 11,
Evaluate g( 3 ). g( 3 ).
Solve g( n )=6. g( n )=6.

n n 1 2 3 4 5
g( n ) g( n ) 8 6 7 6 8
Table 11

Try It #7

Using [link], evaluate g( 1 ). 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 12

Reading Function Values from a Graph

Given the graph in Figure 4,

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

Graph of a positive parabola centered at (1, 0).
Figure 4

Try It #8

Using Figure 4, solve f( x )=1. f( x )=1.

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 12.

Letter grade Grade point average
A 4.0
B 3.0
C 2.0
D 1.0
Table 12

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 Figure 1(a) and Figure 1(b). The function in part (a) shows a relationship that is not a one-to-one function because inputs qq and rr both give output n.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 Function

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

Example 13

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?

Try It #9

  1. Is a balance a function of the bank account number?
  2. Is a bank account number a function of the balance?
  3. Is a balance a one-to-one function of the bank account number?

Try It #10

Evaluate the following:

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

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 xx and the output value y,y, and we say yy is a function of x,x, or y=f( x )y=f( x ) when the function is named f.f. The graph of the function is the set of all points (x,y)(x,y) in the plane that satisfies the equation y=f( x ). 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 7 tell us that f( 0 )=2f( 0 )=2 and f( 6 )=1.f( 6 )=1. However, the set of all points (x,y)(x,y) satisfying y=f( x )y=f( x ) is a curve. The curve shown includes (0,2)(0,2) and (6,1)(6,1) because the curve passes through those points.

Graph of a polynomial.
Figure 7

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 8.

Three graphs visually showing what is and is not a function.
Figure 8

How To

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

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

Example 14

Applying the Vertical Line Test

Which of the graphs in Figure 9 represent(s) a function y=f( x )? y=f( x )?

Graph of a polynomial.
Figure 9

Try It #11

Does the graph in Figure 11 represent a function?

Graph of absolute value function.
Figure 11

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.

How To

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

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

Example 15

Applying the Horizontal Line Test

Consider the functions shown in Figure 9(a) and Figure 9(b). Are either of the functions one-to-one?

Try It #12

Is the graph shown in Figure 9 one-to-one?

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 xx as the input variable and y=f( x )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 13.

Toolkit Functions
Name Function Graph
Constant f( x )=c, f( x )=c, where c c is a constant Graph of a constant function.
Identity f( x )=x f( x )=x Graph of a straight line.
Absolute value f( x )=| x | f( x )=| x | Graph of absolute function.
Quadratic f( x )= x 2 f( x )= x 2 Graph of a parabola.
Cubic f( x )= x 3 f( x )= x 3 Graph of f(x) = x^3.
Reciprocal f( x )= 1 x f( x )= 1 x Graph of f(x)=1/x.
Reciprocal squared f( x )= 1 x 2 f( x )= 1 x 2 Graph of f(x)=1/x^2.
Square root f( x )= x f( x )= x Graph of f(x)=sqrt(x).
Cube root f( x )= x 3 f( x )= x 3 Graph of f(x)=x^(1/3).
Table 13

Media

3.1 Section Exercises

Verbal

1.

What is the difference between a relation and a function?

2.

What is the difference between the input and the output of a function?

3.

Why does the vertical line test tell us whether the graph of a relation represents a function?

4.

How can you determine if a relation is a one-to-one function?

5.

Why does the horizontal line test tell us whether the graph of a function is one-to-one?

Algebraic

For the following exercises, determine whether the relation represents a function.

6.

{ ( a,b ),( c,d ),( a,c ) } { ( a,b ),( c,d ),( a,c ) }

7.

{(a,b),(b,c),(c,c)} {(a,b),(b,c),(c,c)}

For the following exercises, determine whether the relation represents yy as a function of x. x.

8.

5x+2y=10 5x+2y=10

9.

y= x 2 y= x 2

10.

x= y 2 x= y 2

11.

3 x 2 +y=14 3 x 2 +y=14

12.

2x+ y 2 =6 2x+ y 2 =6

13.

y=2 x 2 +40x y=2 x 2 +40x

14.

y= 1 x y= 1 x

15.

x= 3y+5 7y1 x= 3y+5 7y1

16.

x= 1 y 2 x= 1 y 2

17.

y= 3x+5 7x1 y= 3x+5 7x1

18.

x 2 + y 2 =9 x 2 + y 2 =9

19.

2xy=1 2xy=1

20.

x= y 3 x= y 3

21.

y= x 3 y= x 3

22.

y= 1 x 2 y= 1 x 2

23.

x=± 1y x=± 1y

24.

y=± 1x y=± 1x

25.

y 2 = x 2 y 2 = x 2

26.

y 3 = x 2 y 3 = x 2

For the following exercises, evaluate f(−3),f(2),f(a),f(a),f(a+h). f(−3),f(2),f(a),f(a),f(a+h).

27.

f(x)=2x5 f(x)=2x5

28.

f(x)=5 x 2 +2x1 f(x)=5 x 2 +2x1

29.

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

30.

f(x)= 6x1 5x+2 f(x)= 6x1 5x+2

31.

f(x)=| x1 || x+1 | f(x)=| x1 || x+1 |

32.

Given the function g(x)=5 x 2 , g(x)=5 x 2 , evaluate g(x+h)g(x) h ,h0. g(x+h)g(x) h ,h0.

33.

Given the function g(x)= x 2 +2x, g(x)= x 2 +2x, evaluate g(x)g(a) xa ,xa. g(x)g(a) xa ,xa.

34.

Given the function k(t)=2t1:k(t)=2t1:

  1. Evaluate k(2).k(2).
  2. Solve k(t)=7.k(t)=7.
35.

Given the function f(x)=83x:f(x)=83x:

  1. Evaluate f(2).f(2).
  2. Solve f(x)=1.f(x)=1.
36.

Given the function p(c)= c 2 +c:p(c)= c 2 +c:

  1. Evaluate p(3).p(3).
  2. Solve p(c)=2.p(c)=2.
37.

Given the function f(x)= x 2 3x:f(x)= x 2 3x:

  1. Evaluate f(5).f(5).
  2. Solve f(x)=4.f(x)=4.
38.

Given the function f(x)= x+2 :f(x)= x+2 :

  1. Evaluate f(7).f(7).
  2. Solve f(x)=4.f(x)=4.
39.

Consider the relationship 3r+2t=18.3r+2t=18.

  1. Write the relationship as a function r=f(t).r=f(t).
  2. Evaluate f(3).f(3).
  3. Solve f(t)=2.f(t)=2.

Graphical

For the following exercises, use the vertical line test to determine which graphs show relations that are functions.

40.
Graph of relation.
41.
Graph of relation.
42.
Graph of relation.
43.
Graph of relation.
44.
Graph of relation.
45.
Graph of relation.
46.
Graph of relation.
47.
Graph of relation.
48.
Graph of relation.
49.
Graph of relation.
50.
Graph of relation.
51.
Graph of relation.
52.

Given the following graph,

  1. Evaluate f(−1).f(−1).
  2. Solve for f(x)=3.f(x)=3.
Graph of relation.
53.

Given the following graph,

  1. Evaluate f(0).f(0).
  2. Solve for f(x)=−3.f(x)=−3.
Graph of relation.
54.

Given the following graph,

  1. Evaluate f(4).f(4).
  2. Solve for f(x)=1.f(x)=1.
Graph of relation.

For the following exercises, determine if the given graph is a one-to-one function.

55.
Graph of a circle.
56.
Graph of a parabola.
57.
Graph of a rotated cubic function.
58.
Graph of half of 1/x.
59.
Graph of a one-to-one function.

Numeric

For the following exercises, determine whether the relation represents a function.

60.

{ ( −1,−1 ),( −2,−2 ),( −3,−3 ) } { ( −1,−1 ),( −2,−2 ),( −3,−3 ) }

61.

{ ( 3,4 ),( 4,5 ),( 5,6 ) } { ( 3,4 ),( 4,5 ),( 5,6 ) }

62.

{ (2,5),(7,11),(15,8),(7,9) } { (2,5),(7,11),(15,8),(7,9) }

For the following exercises, determine if the relation represented in table form represents yy as a function of x.x.

63.
x x 5 10 15
y y 3 8 14
64.
x x 5 10 15
y y 3 8 8
65.
x x 5 10 10
y y 3 8 14

For the following exercises, use the function ff represented in the table below.

xx 0 1 2 3 4 5 6 7 8 9
f(x)f(x) 74 28 1 53 56 3 36 45 14 47
Table 14
66.

Evaluate f(3).f(3).

67.

Solve f(x)=1.f(x)=1.

For the following exercises, evaluate the function ff at the values f( 2 ),f(1),f(0),f(1), f( 2 ),f(1),f(0),f(1), and f(2).f(2).

68.

f( x )=42x f( x )=42x

69.

f( x )=83x f( x )=83x

70.

f( x )=8 x 2 7x+3 f( x )=8 x 2 7x+3

71.

f( x )=3+ x+3 f( x )=3+ x+3

72.

f(x)= x2 x+3 f(x)= x2 x+3

73.

f( x )= 3 x f( x )= 3 x

For the following exercises, evaluate the expressions, given functions f,g, f,g, and h:h:

f(x)=3x2 f(x)=3x2
g(x)=5 x 2 g(x)=5 x 2
h(x)=2 x 2 +3x1 h(x)=2 x 2 +3x1

74.

3f( 1 )4g( 2 ) 3f( 1 )4g( 2 )

75.

f( 7 3 )h( 2 ) f( 7 3 )h( 2 )

Technology

For the following exercises, graph y= x 2 y= x 2 on the given domain. Determine the corresponding range. Show each graph.

76.

[0.1,0.1] [0.1,0.1]

77.

[10,10] [10,10]

78.

[100,100] [100,100]

For the following exercises, graph y= x 3 y= x 3 on the given domain. Determine the corresponding range. Show each graph.

79.

[0.1,0.1] [0.1,0.1]

80.

[10,10] [10,10]

81.

[100,100] [100,100]

For the following exercises, graph y= x y= x on the given domain. Determine the corresponding range. Show each graph.

82.

[0,0.01] [0,0.01]

83.

[0,100] [0,100]

84.

[0,10,000] [0,10,000]

For the following exercises, graph y= x 3 y= x 3 on the given domain. Determine the corresponding range. Show each graph.

85.

[−0.001,0.001] [−0.001,0.001]

86.

[−1000,1000] [−1000,1000]

87.

[−1,000,000,1,000,000] [−1,000,000,1,000,000]

Real-World Applications

88.

The amount of garbage, G,G, produced by a city with population pp is given by G=f( p ).G=f( p ). G G is measured in tons per week, and pp is measured in thousands of people.

  1. The town of Tola has a population of 40,000 and produces 13 tons of garbage each week. Express this information in terms of the function f.f.
  2. Explain the meaning of the statement f( 5 )=2.f( 5 )=2.
89.

The number of cubic yards of dirt, D,D, needed to cover a garden with area aa square feet is given by D=g( a ).D=g( a ).

  1. A garden with area 5000 ft2 requires 50 yd3 of dirt. Express this information in terms of the function g.g.
  2. Explain the meaning of the statement g( 100 )=1.g( 100 )=1.
90.

Let f( t )f( t ) be the number of ducks in a lake tt years after 1990. Explain the meaning of each statement:

  1. f( 5 )=30 f( 5 )=30
  2. f( 10 )=40 f( 10 )=40
91.

Let h( t )h( t ) be the height above ground, in feet, of a rocket tt seconds after launching. Explain the meaning of each statement:

  1. h( 1 )=200 h( 1 )=200
  2. h( 2 )=350 h( 2 )=350
92.

Show that the function f( x )=3 ( x5 ) 2 +7f( x )=3 ( x5 ) 2 +7 is underlinenotend underline one-to-one.

Footnotes

  • 1http://www.baseball-almanac.com/legendary/lisn100.shtml. Accessed 3/24/2014.
  • 2http://www.kgbanswers.com/how-long-is-a-dogs-memory-span/4221590. Accessed 3/24/2014.
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