Skip to Content
OpenStax Logo
College Algebra

3.4 Composition of Functions

College Algebra3.4 Composition of Functions
Buy book
  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:
  • Combine functions using algebraic operations.
  • Create a new function by composition of functions.
  • Evaluate composite functions.
  • Find the domain of a composite function.
  • Decompose a composite function into its component functions.

Suppose we want to calculate how much it costs to heat a house on a particular day of the year. The cost to heat a house will depend on the average daily temperature, and in turn, the average daily temperature depends on the particular day of the year. Notice how we have just defined two relationships: The cost depends on the temperature, and the temperature depends on the day.

Using descriptive variables, we can notate these two functions. The function C( T ) C( T )gives the cost C Cof heating a house for a given average daily temperature in T Tdegrees Celsius. The function T( d ) T( d )gives the average daily temperature on day d dof the year. For any given day, Cost=C( T( d ) ) Cost=C( T( d ) )means that the cost depends on the temperature, which in turns depends on the day of the year. Thus, we can evaluate the cost function at the temperature T( d ). T( d ).For example, we could evaluate T( 5 ) T( 5 )to determine the average daily temperature on the 5th day of the year. Then, we could evaluate the cost function at that temperature. We would write C( T( 5 ) ). C( T( 5 ) ).

Explanation of C(T(5)), which is the cost for the temperature and T(5) is the temperature on day 5.

By combining these two relationships into one function, we have performed function composition, which is the focus of this section.

Combining Functions Using Algebraic Operations

Function composition is only one way to combine existing functions. Another way is to carry out the usual algebraic operations on functions, such as addition, subtraction, multiplication and division. We do this by performing the operations with the function outputs, defining the result as the output of our new function.

Suppose we need to add two columns of numbers that represent a husband and wife’s separate annual incomes over a period of years, with the result being their total household income. We want to do this for every year, adding only that year’s incomes and then collecting all the data in a new column. If w(y) w(y)is the wife’s income and h(y) h(y)is the husband’s income in year y, y,and we want T Tto represent the total income, then we can define a new function.

T( y )=h( y )+w( y ) T( y )=h( y )+w( y )

If this holds true for every year, then we can focus on the relation between the functions without reference to a year and write

T=h+w T=h+w

Just as for this sum of two functions, we can define difference, product, and ratio functions for any pair of functions that have the same kinds of inputs (not necessarily numbers) and also the same kinds of outputs (which do have to be numbers so that the usual operations of algebra can apply to them, and which also must have the same units or no units when we add and subtract). In this way, we can think of adding, subtracting, multiplying, and dividing functions.

For two functions f( x ) f( x )and g( x ) g( x )with real number outputs, we define new functions f+g,fg,fg, f+g,fg,fg,and f g f g by the relations

(f+g)(x) = f(x)+g(x) (fg)(x) = f(x)g(x)  (fg)(x) = f(x)g(x) ( f g )(x) = f(x) g(x) whereg(x)0 (f+g)(x) = f(x)+g(x) (fg)(x) = f(x)g(x)  (fg)(x) = f(x)g(x) ( f g )(x) = f(x) g(x) whereg(x)0

Example 1

Performing Algebraic Operations on Functions

Find and simplify the functions ( gf )( x ) ( gf )( x )and ( g f )( x ), ( g f )( x ),given f( x )=x1 f( x )=x1and g( x )= x 2 1. g( x )= x 2 1.Are they the same function?

Try It #1

Find and simplify the functions ( fg )( x ) ( fg )( x )and ( fg )( x ). ( fg )( x ).

f( x )=x1    and    g( x )= x 2 1 f( x )=x1    and    g( x )= x 2 1

Are they the same function?

Create a Function by Composition of Functions

Performing algebraic operations on functions combines them into a new function, but we can also create functions by composing functions. When we wanted to compute a heating cost from a day of the year, we created a new function that takes a day as input and yields a cost as output. The process of combining functions so that the output of one function becomes the input of another is known as a composition of functions. The resulting function is known as a composite function. We represent this combination by the following notation:

( fg )( x )=f( g( x ) ) ( fg )( x )=f( g( x ) )

We read the left-hand side as f f composed with g gat x,” x,” and the right-hand side as f f of g gof x. x. The two sides of the equation have the same mathematical meaning and are equal. The open circle symbol is called the composition operator. We use this operator mainly when we wish to emphasize the relationship between the functions themselves without referring to any particular input value. Composition is a binary operation that takes two functions and forms a new function, much as addition or multiplication takes two numbers and gives a new number. However, it is important not to confuse function composition with multiplication because, as we learned above, in most cases f(g(x))f(x)g(x). f(g(x))f(x)g(x).

It is also important to understand the order of operations in evaluating a composite function. We follow the usual convention with parentheses by starting with the innermost parentheses first, and then working to the outside. In the equation above, the function g gtakes the input x xfirst and yields an output g( x ). g( x ).Then the function f ftakes g( x ) g( x )as an input and yields an output f( g( x ) ). f( g( x ) ).

Explanation of the composite function.

In general, fg fgand gf gfare different functions. In other words, in many cases f( g( x ) )g( f( x ) ) f( g( x ) )g( f( x ) )for all x. x.We will also see that sometimes two functions can be composed only in one specific order.

For example, if f( x )= x 2 f( x )= x 2 and g( x )=x+2, g( x )=x+2,then

f(g(x)) = f(x+2) = (x+2) 2 = x 2 +4x+4 f(g(x)) = f(x+2) = (x+2) 2 = x 2 +4x+4

but

g(f(x)) = g( x 2 ) = x 2 +2 g(f(x)) = g( x 2 ) = x 2 +2

These expressions are not equal for all values of x, x,so the two functions are not equal. It is irrelevant that the expressions happen to be equal for the single input value x= 1 2 . x= 1 2 .

Note that the range of the inside function (the first function to be evaluated) needs to be within the domain of the outside function. Less formally, the composition has to make sense in terms of inputs and outputs.

Composition of Functions

When the output of one function is used as the input of another, we call the entire operation a composition of functions. For any input x xand functions f fand g, g,this action defines a composite function, which we write as fg fgsuch that

( fg )( x )=f( g( x ) ) ( fg )( x )=f( g( x ) )

The domain of the composite function fg fgis all x xsuch that x xis in the domain of g gand g( x ) g( x )is in the domain of f. f.

It is important to realize that the product of functions fg fgis not the same as the function composition f( g( x ) ), f( g( x ) ),because, in general, f( x )g( x )f( g( x ) ). f( x )g( x )f( g( x ) ).

Example 2

Determining whether Composition of Functions is Commutative

Using the functions provided, find f( g( x ) ) f( g( x ) )and g( f( x ) ). g( f( x ) ).Determine whether the composition of the functions is commutative.

f(x)=2x+1g(x)=3x f(x)=2x+1g(x)=3x

Example 3

Interpreting Composite Functions

The function c(s) c(s)gives the number of calories burned completing s ssit-ups, and s(t) s(t)gives the number of sit-ups a person can complete in t tminutes. Interpret c(s(3)). c(s(3)).

Example 4

Investigating the Order of Function Composition

Suppose f(x) f(x)gives miles that can be driven in x xhours and g(y) g(y)gives the gallons of gas used in driving y ymiles. Which of these expressions is meaningful: f( g(y) ) f( g(y) )or g( f(x) )? g( f(x) )?

Q&A

Are there any situations where f(g(y)) f(g(y)) and g(f(x)) g(f(x)) would both be meaningful or useful expressions?

Yes. For many pure mathematical functions, both compositions make sense, even though they usually produce different new functions. In real-world problems, functions whose inputs and outputs have the same units also may give compositions that are meaningful in either order.

Try It #2

The gravitational force on a planet a distance r from the sun is given by the function G(r). G(r). The acceleration of a planet subjected to any force F F is given by the function a(F). a(F). Form a meaningful composition of these two functions, and explain what it means.

Evaluating Composite Functions

Once we compose a new function from two existing functions, we need to be able to evaluate it for any input in its domain. We will do this with specific numerical inputs for functions expressed as tables, graphs, and formulas and with variables as inputs to functions expressed as formulas. In each case, we evaluate the inner function using the starting input and then use the inner function’s output as the input for the outer function.

Evaluating Composite Functions Using Tables

When working with functions given as tables, we read input and output values from the table entries and always work from the inside to the outside. We evaluate the inside function first and then use the output of the inside function as the input to the outside function.

Example 5

Using a Table to Evaluate a Composite Function

Using Table 1, evaluate f(g(3)) f(g(3))and g(f(3)). g(f(3)).

x x f(x) f(x) g(x) g(x)
1 6 3
2 8 5
3 3 2
4 1 7
Table 1

Try It #3

Using Table 1, evaluate f(g(1)) f(g(1))and g(f(4)). g(f(4)).

Evaluating Composite Functions Using Graphs

When we are given individual functions as graphs, the procedure for evaluating composite functions is similar to the process we use for evaluating tables. We read the input and output values, but this time, from the x- x- and y- y-axes of the graphs.

How To

Given a composite function and graphs of its individual functions, evaluate it using the information provided by the graphs.

  1. Locate the given input to the inner function on the x- x-axis of its graph.
  2. Read off the output of the inner function from the y- y-axis of its graph.
  3. Locate the inner function output on the x- x-axis of the graph of the outer function.
  4. Read the output of the outer function from the y- y-axis of its graph. This is the output of the composite function.

Example 6

Using a Graph to Evaluate a Composite Function

Using Figure 1, evaluate f(g(1)). f(g(1)).

Explanation of the composite function.
Figure 1

Analysis

Figure 3 shows how we can mark the graphs with arrows to trace the path from the input value to the output value.

Two graphs of a positive and negative parabola.
Figure 3

Try It #4

Using Figure 1, evaluate g(f(2)). g(f(2)).

Evaluating Composite Functions Using Formulas

When evaluating a composite function where we have either created or been given formulas, the rule of working from the inside out remains the same. The input value to the outer function will be the output of the inner function, which may be a numerical value, a variable name, or a more complicated expression.

While we can compose the functions for each individual input value, it is sometimes helpful to find a single formula that will calculate the result of a composition f( g( x ) ). f( g( x ) ).To do this, we will extend our idea of function evaluation. Recall that, when we evaluate a function like f(t)= t 2 t, f(t)= t 2 t,we substitute the value inside the parentheses into the formula wherever we see the input variable.

How To

Given a formula for a composite function, evaluate the function.

  1. Evaluate the inside function using the input value or variable provided.
  2. Use the resulting output as the input to the outside function.

Example 7

Evaluating a Composition of Functions Expressed as Formulas with a Numerical Input

Given f(t)= t 2 t f(t)= t 2 tand h(x)=3x+2, h(x)=3x+2,evaluate f(h(1)). f(h(1)).

Analysis

It makes no difference what the input variables t tand x xwere called in this problem because we evaluated for specific numerical values.

Try It #5

Given f(t)= t 2 t f(t)= t 2 tand h(x)=3x+2, h(x)=3x+2,evaluate

  1. h(f(2)) h(f(2))
  2. h(f(2)) h(f(2))

Finding the Domain of a Composite Function

As we discussed previously, the domain of a composite function such as fg fgis dependent on the domain of g gand the domain of f. f.It is important to know when we can apply a composite function and when we cannot, that is, to know the domain of a function such as fg. fg.Let us assume we know the domains of the functions f fand g gseparately. If we write the composite function for an input x xas f( g( x ) ), f( g( x ) ),we can see right away that x xmust be a member of the domain of g gin order for the expression to be meaningful, because otherwise we cannot complete the inner function evaluation. However, we also see that g( x ) g( x )must be a member of the domain of f, f,otherwise the second function evaluation in f( g( x ) ) f( g( x ) )cannot be completed, and the expression is still undefined. Thus the domain of fg fgconsists of only those inputs in the domain of g gthat produce outputs from g gbelonging to the domain of f. f.Note that the domain of f fcomposed with g gis the set of all x xsuch that x xis in the domain of g gand g( x ) g( x )is in the domain of f. f.

Domain of a Composite Function

The domain of a composite function f( g( x ) ) f( g( x ) )is the set of those inputs x xin the domain of g gfor which g( x ) g( x )is in the domain of f. f.

How To

Given a function composition f(g(x)), f(g(x)), determine its domain.

  1. Find the domain of g. g.
  2. Find the domain of f. f.
  3. Find those inputs x xin the domain of g gfor which g( x ) g( x )is in the domain of f. f.That is, exclude those inputs x xfrom the domain of g gfor which g( x ) g( x )is not in the domain of f. f.The resulting set is the domain of fg. fg.

Example 8

Finding the Domain of a Composite Function

Find the domain of

( fg )(x)    wheref(x)= 5 x1   and g(x)= 4 3x2 ( fg )(x)    wheref(x)= 5 x1   and g(x)= 4 3x2

Example 9

Finding the Domain of a Composite Function Involving Radicals

Find the domain of

( fg )(x)  wheref(x)= x+2   and  g(x)= 3x ( fg )(x)  wheref(x)= x+2   and  g(x)= 3x

Analysis

This example shows that knowledge of the range of functions (specifically the inner function) can also be helpful in finding the domain of a composite function. It also shows that the domain of fg fgcan contain values that are not in the domain of f, f,though they must be in the domain of g. g.

Try It #6

Find the domain of

( fg )(x)  wheref(x)= 1 x2   and  g(x)= x+4 ( fg )(x)  wheref(x)= 1 x2   and  g(x)= x+4

Decomposing a Composite Function into its Component Functions

In some cases, it is necessary to decompose a complicated function. In other words, we can write it as a composition of two simpler functions. There may be more than one way to decompose a composite function, so we may choose the decomposition that appears to be most expedient.

Example 10

Decomposing a Function

Write f(x)= 5 x 2 f(x)= 5 x 2 as the composition of two functions.

Try It #7

Write f(x)= 4 3 4+ x 2 f(x)= 4 3 4+ x 2 as the composition of two functions.

Media

Access these online resources for additional instruction and practice with composite functions.

3.4 Section Exercises

Verbal

1.

How does one find the domain of the quotient of two functions, f g ? f g ?

2.

What is the composition of two functions, fg? fg?

3.

If the order is reversed when composing two functions, can the result ever be the same as the answer in the original order of the composition? If yes, give an example. If no, explain why not.

4.

How do you find the domain for the composition of two functions, fg? fg?

Algebraic

For the following exercises, determine the domain for each function in interval notation.

5.

Given f(x)= x 2 +2x  f(x)= x 2 +2x  and  g(x)=6 x 2 ,  g(x)=6 x 2 , find f+g,fg,fg,  f+g,fg,fg,  and   f g .   f g .

6.

Given f(x)=3 x 2 +x  f(x)=3 x 2 +x  and  g(x)=5,  g(x)=5, find f+g,fg,fg, f+g,fg,fg, and   f g .   f g .

7.

Given f(x)=2 x 2 +4x  f(x)=2 x 2 +4x  and  g(x)= 1 2x ,  g(x)= 1 2x , find f+g,fg,fg,  f+g,fg,fg,  and   f g .   f g .

8.

Given f(x)= 1 x4 f(x)= 1 x4 and g(x)= 1 6x , g(x)= 1 6x ,find f+g,fg,fg,  f+g,fg,fg,  and   f g .   f g .

9.

Given f(x)=3 x 2 f(x)=3 x 2 and g(x)= x5 , g(x)= x5 ,find f+g,fg,fg,  f+g,fg,fg,  and   f g .   f g .

10.

Given f(x)= x f(x)= x and g(x)=|x3|, g(x)=|x3|,find g f . g f .

11.

For the following exercise, find the indicated function given f(x)=2 x 2 +1 f(x)=2 x 2 +1and g(x)=3x5. g(x)=3x5.

  1. f(g(2)) f(g(2))
  2. f(g(x)) f(g(x))
  3. g(f(x)) g(f(x))
  4. ( gg )( x ) ( gg )( x )
  5. ( ff )( 2 ) ( ff )( 2 )

For the following exercises, use each pair of functions to find f( g( x ) ) f( g( x ) )and g( f( x ) ). g( f( x ) ).Simplify your answers.

12.

f(x)= x 2 +1,g(x)= x+2 f(x)= x 2 +1,g(x)= x+2

13.

f(x)= x +2,g(x)= x 2 +3 f(x)= x +2,g(x)= x 2 +3

14.

f(x)=| x |,g(x)=5x+1 f(x)=| x |,g(x)=5x+1

15.

f(x)= x 3 ,g(x)= x+1 x 3 f(x)= x 3 ,g(x)= x+1 x 3

16.

f(x)= 1 x6 ,g(x)= 7 x +6 f(x)= 1 x6 ,g(x)= 7 x +6

17.

f(x)= 1 x4 ,g(x)= 2 x +4 f(x)= 1 x4 ,g(x)= 2 x +4

For the following exercises, use each set of functions to find f( g( h(x) ) ). f( g( h(x) ) ).Simplify your answers.

18.

f(x)= x 4 +6, f(x)= x 4 +6, g(x)=x6, g(x)=x6,and h(x)= x h(x)= x

19.

f(x)= x 2 +1, f(x)= x 2 +1, g(x)= 1 x , g(x)= 1 x ,and h(x)=x+3 h(x)=x+3

20.

Given f(x)= 1 x f(x)= 1 x and g(x)=x3, g(x)=x3,find the following:

  1. (fg)(x) (fg)(x)
  2. the domain of (fg)(x) (fg)(x)in interval notation
  3. (gf)(x) (gf)(x)
  4. the domain of (gf)(x) (gf)(x)
  5. ( f g )(x) ( f g )(x)
21.

Given f(x)= 24x f(x)= 24x and g(x)= 3 x , g(x)= 3 x ,find the following:

  1. (gf)(x) (gf)(x)
  2. the domain of (gf)(x) (gf)(x)in interval notation
22.

Given the functions f(x)= 1x x andg(x)= 1 1+ x 2 , f(x)= 1x x andg(x)= 1 1+ x 2 , find the following:

  1. (gf)(x) (gf)(x)
  2. (gf)(2) (gf)(2)
23.

Given functions p(x)= 1 x p(x)= 1 x and m(x)= x 2 4, m(x)= x 2 4,state the domain of each of the following functions using interval notation:

  1. p(x) m(x) p(x) m(x)
  2. p(m(x)) p(m(x))
  3. m(p(x)) m(p(x))
24.

Given functions q(x)= 1 x q(x)= 1 x and h(x)= x 2 9, h(x)= x 2 9,state the domain of each of the following functions using interval notation.

  1. q(x) h(x) q(x) h(x)
  2. q( h(x) ) q( h(x) )
  3. h( q(x) ) h( q(x) )
25.

For f(x)= 1 x f(x)= 1 x and g(x)= x1 , g(x)= x1 ,write the domain of (fg)(x) (fg)(x)in interval notation.

For the following exercises, find functions f(x) f(x)and g(x) g(x)so the given function can be expressed as h(x)=f( g(x) ). h(x)=f( g(x) ).

26.

h(x)= (x+2) 2 h(x)= (x+2) 2

27.

h(x)= (x5) 3 h(x)= (x5) 3

28.

h(x)= 3 x5 h(x)= 3 x5

29.

h(x)= 4 (x+2) 2 h(x)= 4 (x+2) 2

30.

h(x)=4+ x 3 h(x)=4+ x 3

31.

h(x)= 1 2x3 3 h(x)= 1 2x3 3

32.

h(x)= 1 (3 x 2 4) 3 h(x)= 1 (3 x 2 4) 3

33.

h(x)= 3x2 x+5 4 h(x)= 3x2 x+5 4

34.

h(x)= ( 8+ x 3 8 x 3 ) 4 h(x)= ( 8+ x 3 8 x 3 ) 4

35.

h(x)= 2x+6 h(x)= 2x+6

36.

h(x)= (5x1) 3 h(x)= (5x1) 3

37.

h(x)= x1 3 h(x)= x1 3

38.

h(x)=| x 2 +7 | h(x)=| x 2 +7 |

39.

h(x)= 1 (x2) 3 h(x)= 1 (x2) 3

40.

h(x)= ( 1 2x3 ) 2 h(x)= ( 1 2x3 ) 2

41.

h(x)= 2x1 3x+4 h(x)= 2x1 3x+4

Graphical

For the following exercises, use the graphs of f, f,shown in Figure 4, and g, g,shown in Figure 5, to evaluate the expressions.

Graph of a function.
Figure 4
Graph of a function.
Figure 5
42.

f( g(3) ) f( g(3) )

43.

f( g(1) ) f( g(1) )

44.

g( f(1) ) g( f(1) )

45.

g( f(0) ) g( f(0) )

46.

f( f(5) ) f( f(5) )

47.

f( f(4) ) f( f(4) )

48.

g( g(2) ) g( g(2) )

49.

g( g(0) ) g( g(0) )

For the following exercises, use graphs of f(x), f(x),shown in Figure 6, g(x), g(x),shown in Figure 7, and h(x), h(x),shown in Figure 8, to evaluate the expressions.

Figure 6
Graph of a parabola.
Figure 7
Graph of a square root function.
Figure 8
50.

g( f( 1 ) ) g( f( 1 ) )

51.

g( f( 2 ) ) g( f( 2 ) )

52.

f( g( 4 ) ) f( g( 4 ) )

53.

f( g( 1 ) ) f( g( 1 ) )

54.

f( h( 2 ) ) f( h( 2 ) )

55.

h( f( 2 ) ) h( f( 2 ) )

56.

f( g( h( 4 ) ) ) f( g( h( 4 ) ) )

57.

f( g( f( 2 ) ) ) f( g( f( 2 ) ) )

Numeric

For the following exercises, use the function values for f and g f and gshown in Table 3 to evaluate each expression.

x x f(x) f(x) g(x) g(x)
079
165
256
382
441
508
627
713
894
930
Table 3
58.

f( g( 8 ) ) f( g( 8 ) )

59.

f( g( 5 ) ) f( g( 5 ) )

60.

g( f( 5 ) ) g( f( 5 ) )

61.

g( f( 3 ) ) g( f( 3 ) )

62.

f( f( 4 ) ) f( f( 4 ) )

63.

f( f( 1 ) ) f( f( 1 ) )

64.

g( g( 2 ) ) g( g( 2 ) )

65.

g( g( 6 ) ) g( g( 6 ) )

For the following exercises, use the function values for f and g f and gshown in Table 4 to evaluate the expressions.

x x f(x) f(x) g(x) g(x)
−3 −3 11 −8 −8
−2 −2 9 −3 −3
−1 −1 7 0
0 5 1
1 3 0
2 1 −3 −3
3 −1 −1 −8 −8
Table 4
66.

(fg)(1) (fg)(1)

67.

(fg)(2) (fg)(2)

68.

(gf)(2) (gf)(2)

69.

(gf)(3) (gf)(3)

70.

(gg)(1) (gg)(1)

71.

(ff)(3) (ff)(3)

For the following exercises, use each pair of functions to find f( g( 0 ) ) f( g( 0 ) )and g( f(0) ). g( f(0) ).

72.

f(x)=4x+8,g(x)=7 x 2 f(x)=4x+8,g(x)=7 x 2

73.

f(x)=5x+7,g(x)=42 x 2 f(x)=5x+7,g(x)=42 x 2

74.

f(x)= x+4 ,g(x)=12 x 3 f(x)= x+4 ,g(x)=12 x 3

75.

f(x)= 1 x+2 ,g(x)=4x+3 f(x)= 1 x+2 ,g(x)=4x+3

For the following exercises, use the functions f(x)=2 x 2 +1 f(x)=2 x 2 +1and g(x)=3x+5 g(x)=3x+5 to evaluate or find the composite function as indicated.

76.

f( g(2) ) f( g(2) )

77.

f( g(x) ) f( g(x) )

78.

g( f(3) ) g( f(3) )

79.

(gg)(x) (gg)(x)

Extensions

For the following exercises, use f(x)= x 3 +1 f(x)= x 3 +1and g(x)= x1 3 . g(x)= x1 3 .

80.

Find (fg)(x) (fg)(x)and (gf)(x). (gf)(x).Compare the two answers.

81.

Find (fg)(2) (fg)(2)and (gf)(2). (gf)(2).

82.

What is the domain of (gf)(x)? (gf)(x)?

83.

What is the domain of (fg)(x)? (fg)(x)?

84.

Let f(x)= 1 x . f(x)= 1 x .

  1. Find (ff)(x). (ff)(x).
  2. Is (ff)(x) (ff)(x)for any function f fthe same result as the answer to part (a) for any function? Explain.

For the following exercises, let F(x)= (x+1) 5 , F(x)= (x+1) 5 , f(x)= x 5 , f(x)= x 5 ,and g(x)=x+1. g(x)=x+1.

85.

True or False: (gf)(x)=F(x). (gf)(x)=F(x).

86.

True or False: (fg)(x)=F(x). (fg)(x)=F(x).

For the following exercises, find the composition when f(x)= x 2 +2 f(x)= x 2 +2for all x0 x0and g(x)= x2 . g(x)= x2 .

87.

(fg)(6);(gf)(6) (fg)(6);(gf)(6)

88.

(gf)(a);(fg)(a) (gf)(a);(fg)(a)

89.

(fg)(11);(gf)(11) (fg)(11);(gf)(11)

Real-World Applications

90.

The function D(p) D(p)gives the number of items that will be demanded when the price is p. p.The production cost C(x) C(x)is the cost of producing x xitems. To determine the cost of production when the price is $6, you would do which of the following?

  1. Evaluate D( C(6) ). D( C(6) ).
  2. Evaluate C( D(6) ). C( D(6) ).
  3. Solve D( C(x) )=6. D( C(x) )=6.
  4. Solve C( D(p) )=6. C( D(p) )=6.
91.

The function A(d) A(d)gives the pain level on a scale of 0 to 10 experienced by a patient with d dmilligrams of a pain-reducing drug in her system. The milligrams of the drug in the patient’s system after t tminutes is modeled by m(t). m(t).Which of the following would you do in order to determine when the patient will be at a pain level of 4?

  1. Evaluate A( m(4) ). A( m(4) ).
  2. Evaluate m( A(4) ). m( A(4) ).
  3. Solve A( m(t) )=4. A( m(t) )=4.
  4. Solve m( A(d) )=4. m( A(d) )=4.
92.

A store offers customers a 30% discount on the price x xof selected items. Then, the store takes off an additional 15% at the cash register. Write a price function P(x) P(x)that computes the final price of the item in terms of the original price x. x.(Hint: Use function composition to find your answer.)

93.

A rain drop hitting a lake makes a circular ripple. If the radius, in inches, grows as a function of time in minutes according to r(t)=25 t+2 , r(t)=25 t+2 ,find the area of the ripple as a function of time. Find the area of the ripple at t=2. t=2.

94.

A forest fire leaves behind an area of grass burned in an expanding circular pattern. If the radius of the circle of burning grass is increasing with time according to the formula r(t)=2t+1, r(t)=2t+1,express the area burned as a function of time, t t(minutes).

95.

Use the function you found in the previous exercise to find the total area burned after 5 minutes.

96.

The radius r, r,in inches, of a spherical balloon is related to the volume, V, V,by r(V)= 3V 4π 3 . r(V)= 3V 4π 3 .Air is pumped into the balloon, so the volume after t tseconds is given by V(t)=10+20t. V(t)=10+20t.

  1. Find the composite function r( V(t) ). r( V(t) ).
  2. Find the exact time when the radius reaches 10 inches.
97.

The number of bacteria in a refrigerated food product is given by N(T)=23 T 2 56T+1, N(T)=23 T 2 56T+1, 3<T<33, 3<T<33,where T T is the temperature of the food. When the food is removed from the refrigerator, the temperature is given by T(t)=5t+1.5, T(t)=5t+1.5, where t t is the time in hours.

  1. Find the composite function N( T(t) ). N( T(t) ).
  2. Find the time (round to two decimal places) when the bacteria count reaches 6752.
Citation/Attribution

Want to cite, share, or modify this book? This book is Creative Commons Attribution License 4.0 and you must attribute OpenStax.

Attribution information
  • If you are redistributing all or part of this book in a print format, then you must include on every physical page the following attribution:
    Access for free at https://openstax.org/books/college-algebra/pages/1-introduction-to-prerequisites
  • If you are redistributing all or part of this book in a digital format, then you must include on every digital page view the following attribution:
    Access for free at https://openstax.org/books/college-algebra/pages/1-introduction-to-prerequisites
Citation information

© Feb 10, 2020 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License 4.0 license. The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.