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7.4 Function Operations and Composition of Functions

Your NotesVOCABULARY

Composition of functions Composing a function f with a function g is accomplished by replacing each variable in f with the expression for g

OPERATIONS ON FUNCTIONS

Let f(x) and g(x) be any two functions. You can add, subtract, multiply, or divide f(x) and g(x) to form a new function h(x).

Operation Definition Example: f(x) 5 3x, g(x) 5 x 1 3

Addition h(x) 5 f(x) 1 g(x) h(x) 5 3x 1 (x 1 3)5 4x 1 3

Subtraction h(x) 5 f(x) 2 g(x) h(x) 5 3x 2 (x 1 3)5 2x 2 3

Multiplication h(x) 5 f(x) p g(x) h(x) 5 3x(x 1 3) 5 3x2 9x

Division h(x) 5 f(x)

} g(x) h( x) 5 3x } x 1 3 ,

x 23

The domain of h consists of the x-values that are in the domains of both f and g . When h involves division, the domain does not include x-values for which the denominator is equal to zero .

Goal p Perform operations with functions, including composition of functions.

Copyright McDougal Littell/Houghton Mifflin Company Lesson 7.4 Algebra 2 C&S Notetaking Guide 151

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Your Notes


Example 1 Add and Subtract Functions

Let f(x) 5 3x2 1 1 and g(x) 5 x 2 5. Find h(x) and state its domain.

a. h(x) 5 f(x) 1 g(x) b. h(x) 5 f(x) 2 g(x)

Solutiona. f(x) 1 g(x) 5 3x2 1 1 1 (x 2 5)

5 3x2 1 x 2 4

b. f(x) 2 g(x) 5 3x2 1 1 2 (x 2 5)

5 3x2 2 x 1 6

In both parts (a) and (b), the domains of f and g are all real numbers . So, the domain of h is all real numbers .

Example 2 Multiply and Divide Functions

Let f(x) 5 x3 and g(x) 5 7x. Find h(x) and state its domain.

a. h(x) 5 f(x) p g(x) b. h(x) 5 f(x)

} g(x)

Solutiona. f(x) p g(x) 5 (x3)(7x) 5 7x4

The domains of both f and g are all real numbers . So, the domain of h(x) is all real numbers .

b. f(x)

} g(x) 5 x3

} 7x

5 1 } 7 x( 3 2 1 )

5 1 } 7 x 2

Because g(0) 5 0 , f } g is undefined when x 5 0 .

So, the domain of h(x) is all real numbers except 0 .


Your Notes

1. Let f(x) 5 5x3 and g(x) 5 22x3. Find (a) f 1 g,

(b) f 2 g, (c) f p g, (d) f } g , and (e) the domains.

a. 3x3 b. 7x3 c. 210x6 d. 2 1 } 7

e. The domain of f 1 g, f 2 g, and f p g is all real

numbers. The domain of f } g is all real numbers except 0.

Checkpoint Complete the following exercise.

COMPOSITION OF FUNCTIONS

The composition of a function f with a function g is:

h(x) 5 g(f (x))

The domain of h is the set of all x-values where x is in the domain of g and g(x) is in the domain of f .

Inputof g

Domain of g Range of g

Domain of f Range of f

Outputof g

Inputof f

g(x)

Outputof f

f (g(x))x



Your Notes

Homework

Let f (x) 5 6x and g(x) 5 3x 1 5. Find the following.

a. f (g(x)) b. g(f (x))

c. the domain of each composition

SolutionTo find the compositions, write the composition, substitute the expression for the inner function in the outer function, and simplify.

a. f (g(x)) 5 f( 3x 1 5 ) 5 6( 3x 1 5 ) 5 18x 1 30

b. g(f (x)) 5 g( 3x ) 5 3( 3x ) 1 5 5 18x 1 5

c. The domain of each function is all real numbers , so the domain of each composition is all real numbers .

Example 3 Write a Composition of Functions

Let f(x) 5 6x and g(x) 5 3x 1 5. Evaluate f(g(22)).

SolutionTo evaluate f(g(22)), first find g(22):

g(22) 5 3( 22 ) 1 5 5 26 1 5 5 21

Then substitute g(22) 5 21 into f (g(22)):

f(g(22)) 5 f( 21 ) 5 6( 21 ) 5 26

Example 4 Evaluate a Composition of Functions

2. Let f (x) 5 5x 2 4 and g(x) 5 3x. Find (a) f (g(x)), (b) g(f (x)), (c) f (g(23)), and (d) g(f (4)).

a. 15x 2 4 b. 15x 2 12 c. 249 d. 48

Checkpoint Complete the following exercise.


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