For Problems 31 − 40 , find the inverse of the given function by using the process illustrated in Examples 3 and 4 of this section, and then verify that ( f ∘ f − 1 ) ( x ) = x and ( f − 1 ∘ f ) ( x ) = x (Objective 4) f ( x ) = − 6 x + 2

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To determine To find: The inverse of the given function f ( x ) = − 6 x + 2 and verify ( f ∘ f − 1 ) ( x ) = x and ( f − 1 ∘ f ) ( x ) = x . Explanation Given: The given function is, f ( x ) = − 6 x + 2 Approach: The inverse of a function f is denoted by f − 1 . If ( a , b ) is a ordered pair of f then ( b , a ) is an ordered pair of f − 1 . The domain and range of f − 1 are the range and domain respectively of f . The compositions of function and inverse of the same function is ( f ∘ f − 1 ) ( x ) = f ( f − 1 ( x ) ) , for all x in the domain of f − 1 such that f − 1 ( x ) is the domain of f and ( f − 1 ∘ f ) ( x ) = f − 1 ( f ( x ) ) , for all x in the domain of f such that f ( x ) is the domain of f − 1 . Calculation: The function is given by, f ( x ) = − 6 x + 2 ... ( 1 ) Substitute y for f ( x ) in equation ( 1 ) . y = − 6 x + 2 ... ( 2 ) Substitute x for y in equation ( 2 ) . x = − 6 y + 2 y = x − 2 − 6 ... ( 3 ) Substitute f − 1 ( x ) for y in equation ( 3 ) . f − 1 ( x ) = x − 2 − 6 ... ( 4 ) Therefore, the inverse of the given function is x − 2 − 6 . The compositions of function and inverse of function is obtained by, ( f ∘ f − 1 ) ( x ) = f ( f − 1 ( x ) ) ... ( 5 ) ( f − 1 ∘ f ) ( x ) = f − 1 ( f ( x ) )

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Answer 1

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Intermediate Algebra

10th Edition

Jerome E. Kaufmann + 1 other

Publisher: Cengage Learning

ISBN: 9781285195728

Solutions

Chapter

Section

Answer 2

BuyFind

Intermediate Algebra

10th Edition

Jerome E. Kaufmann + 1 other

Publisher: Cengage Learning

ISBN: 9781285195728

Solutions

Chapter

Section

Answer 3

Solutions

Chapter

Section

Answer 4

Chapter

Section

Answer 5

To determine

To find:

The inverse of the given function f(x)=−6x+2 and verify (f∘f−1)(x)=x and (f−1∘f)(x)=x.

Explanation of Solution

Given:

The given function is,

f(x)=−6x+2

Approach:

The inverse of a function f is denoted by f−1. If (a,b) is a ordered pair of f then (b,a) is an ordered pair of f−1. The domain and range of f−1 are the range and domain respectively of f.

The compositions of function and inverse of the same function is (f∘f−1)(x)=f(f−1(x)), for all x in the domain of f−1 such that f−1(x) is the domain of f and (f−1∘f)(x)=f−1(f(x)), for all x in the domain of f such that f(x) is the domain of f−1.

Calculation:

The function is given by,

f(x)=−6x+2 ...(1)

Substitute y for f(x) in equation (1).

y=−6x+2 ...(2)

Substitute x for y in equation (2).

x=−6y+2y=x−2−6 ...(3)

Substitute f−1(x) for y in equation (3).

f−1(x)=x−2−6 ...(4)

Therefore, the inverse of the given function is x−2−6.

The compositions of function and inverse of function is obtained by,

(f∘f−1)(x)=f(f−1(x)) ...(5)

(f−1∘f)(x)=f−1(f(x))

For Problems 31 − 40 , find the inverse of the given function by using the process illustrated in Examples 3 and 4 of this section, and then verify that ( f ∘ f − 1 ) ( x ) = x and ( f − 1 ∘ f ) ( x ) = x (Objective 4) f ( x ) = − 6 x + 2

Intermediate Algebra

Solutions

Intermediate Algebra

For Problems 31 − 40 , find the inverse of the given function by using the process illustrated in Examples 3 and 4 of this section, and then verify that ( f ∘ f − 1 ) ( x ) = x and ( f − 1 ∘ f ) ( x ) = x (Objective 4)

f ( x ) = − 6 x + 2

Explanation of Solution

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