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College Algebra

3.1 Functions and Function Notation

College Algebra3.1 Functions and Function Notation
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  1. Preface
  2. 1 Prerequisites
    1. Introduction to Prerequisites
    2. 1.1 Real Numbers: Algebra Essentials
    3. 1.2 Exponents and Scientific Notation
    4. 1.3 Radicals and Rational Exponents
    5. 1.4 Polynomials
    6. 1.5 Factoring Polynomials
    7. 1.6 Rational Expressions
    8. Key Terms
    9. Key Equations
    10. Key Concepts
    11. Review Exercises
    12. Practice Test
  3. 2 Equations and Inequalities
    1. Introduction to Equations and Inequalities
    2. 2.1 The Rectangular Coordinate Systems and Graphs
    3. 2.2 Linear Equations in One Variable
    4. 2.3 Models and Applications
    5. 2.4 Complex Numbers
    6. 2.5 Quadratic Equations
    7. 2.6 Other Types of Equations
    8. 2.7 Linear Inequalities and Absolute Value Inequalities
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Review Exercises
    13. Practice Test
  4. 3 Functions
    1. Introduction to Functions
    2. 3.1 Functions and Function Notation
    3. 3.2 Domain and Range
    4. 3.3 Rates of Change and Behavior of Graphs
    5. 3.4 Composition of Functions
    6. 3.5 Transformation of Functions
    7. 3.6 Absolute Value Functions
    8. 3.7 Inverse Functions
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Review Exercises
    13. Practice Test
  5. 4 Linear Functions
    1. Introduction to Linear Functions
    2. 4.1 Linear Functions
    3. 4.2 Modeling with Linear Functions
    4. 4.3 Fitting Linear Models to Data
    5. Key Terms
    6. Key Concepts
    7. Review Exercises
    8. Practice Test
  6. 5 Polynomial and Rational Functions
    1. Introduction to Polynomial and Rational Functions
    2. 5.1 Quadratic Functions
    3. 5.2 Power Functions and Polynomial Functions
    4. 5.3 Graphs of Polynomial Functions
    5. 5.4 Dividing Polynomials
    6. 5.5 Zeros of Polynomial Functions
    7. 5.6 Rational Functions
    8. 5.7 Inverses and Radical Functions
    9. 5.8 Modeling Using Variation
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. Practice Test
  7. 6 Exponential and Logarithmic Functions
    1. Introduction to Exponential and Logarithmic Functions
    2. 6.1 Exponential Functions
    3. 6.2 Graphs of Exponential Functions
    4. 6.3 Logarithmic Functions
    5. 6.4 Graphs of Logarithmic Functions
    6. 6.5 Logarithmic Properties
    7. 6.6 Exponential and Logarithmic Equations
    8. 6.7 Exponential and Logarithmic Models
    9. 6.8 Fitting Exponential Models to Data
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. Practice Test
  8. 7 Systems of Equations and Inequalities
    1. Introduction to Systems of Equations and Inequalities
    2. 7.1 Systems of Linear Equations: Two Variables
    3. 7.2 Systems of Linear Equations: Three Variables
    4. 7.3 Systems of Nonlinear Equations and Inequalities: Two Variables
    5. 7.4 Partial Fractions
    6. 7.5 Matrices and Matrix Operations
    7. 7.6 Solving Systems with Gaussian Elimination
    8. 7.7 Solving Systems with Inverses
    9. 7.8 Solving Systems with Cramer's Rule
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. Practice Test
  9. 8 Analytic Geometry
    1. Introduction to Analytic Geometry
    2. 8.1 The Ellipse
    3. 8.2 The Hyperbola
    4. 8.3 The Parabola
    5. 8.4 Rotation of Axes
    6. 8.5 Conic Sections in Polar Coordinates
    7. Key Terms
    8. Key Equations
    9. Key Concepts
    10. Review Exercises
    11. Practice Test
  10. 9 Sequences, Probability, and Counting Theory
    1. Introduction to Sequences, Probability and Counting Theory
    2. 9.1 Sequences and Their Notations
    3. 9.2 Arithmetic Sequences
    4. 9.3 Geometric Sequences
    5. 9.4 Series and Their Notations
    6. 9.5 Counting Principles
    7. 9.6 Binomial Theorem
    8. 9.7 Probability
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Review Exercises
    13. Practice Test
  11. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
  12. Index

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.

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 consisting of the first components of each ordered pair is called the domain and the set consisting 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 fis a relation that assigns a single element in the range to each element 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 domain, {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 q qand r rboth 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 qis 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 in Figure 2 consists of items and their prices.

  1. Is price a function of the item?
  2. 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.
Figure 2

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 graphing calculators and 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.

h is f of 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 “f of a.” h is f of 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 “f of a.”

Remember, we can use any letter to name the function; the notation h( a ) h( a )shows us that h hdepends on a. a.The value a amust be put into the function h hto 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 y yis 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 )=300 f( 2005 )=300represent?

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 “yy is a function of xx?”

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 y ywe get by applying f fto 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 f fwhere 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 g gis the name of the function that takes the input n nand 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. Replace 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 +3x4 f( x )= x 2 +3x4at

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

For part (d), valuate the function at the indicated values and calculate the quotient.

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=12 2n+6p=12expresses a functional relationship between n n and p. p. We can rewrite it to decide if p pis 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=12 2n+6p=12as a function p=f( n ), p=f( n ),if possible.

Example 10

Expressing the Equation of a Circle as a Function

Does the equation x 2 + y 2 =1 x 2 + y 2 =1represent a function with x xas input and y yas 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 y yas a function of x. x.

Q&A

Are there relationships expressed by an equation that do represent a function but that 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 y yas a function of x, x,there is no simple algebraic formula involving only x xthat equals y. y.However, each x xdoes determine a unique value for y, y,and there are mathematical procedures by which y ycan be found to any desired accuracy. In this case, we say that the equation gives an implicit (implied) rule for y yas 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 P Pat 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 P Pseems 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,

  1. Evaluate g( 3 ). g( 3 ).
  2. 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 Table 11, 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 6,

  1. Evaluate f( 2 ). f( 2 ).
  2. Solve f( x )=4. f( x )=4.
Graph of a positive parabola centered at (1, 0).
Figure 6

Try It #8

Using Figure 6, 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 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 q qand r rboth 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. There are no repeated x- or y-values.

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
  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 x xand the output value y, y,and we say y yis 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 consists of 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 9 tell us that f( 0 )=2 f( 0 )=2and 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 9

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 10.

Three graphs visually showing what is and is not a function.
Figure 10

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 11 represent(s) a function y=f( x )? y=f( x )?

Graph of a polynomial.
Figure 11
Try It #11

Does the graph in Figure 13 represent a function?

Graph of absolute value function.
Figure 13

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 11(a) and Figure 11(b). Are either of the functions one-to-one?

Try It #12

Is the graph shown in Figure 12 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 x xas 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

Footnotes

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

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 y yas 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 the function f fat the indicated values  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 ,simplify 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,simplify 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,

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

Given the following graph,

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

Given the following graph,

  • Evaluate f(4). f(4).
  • 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 y yas 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 f frepresented in Table 14.

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

Evaluate f(3). f(3).

67.

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

For the following exercises, evaluate the function f fat 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 viewing window. Determine the corresponding range for each viewing window. 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 viewing window. Determine the corresponding range for each viewing window. 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 viewing window. Determine the corresponding range for each viewing window. 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 viewing window. Determine the corresponding range for each viewing window. 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 p pis given by G=f( p ). G=f( p ). G Gis measured in tons per week, and p pis 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 a asquare 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 t tyears 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 t tseconds 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 +7 f( x )=3 ( x5 ) 2 +7is not one-to-one.

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