Determine the inverse for each - then confirm, verify, and prove the inverse.

Anonymous

Question Description

Determine the inverse for each - then confirm, verify, and prove the inverse.

I need 5 problems C, K, O, R and S worked out please according to the above directions.

Determine the inverse for each - then  confirm, verify, and prove the inverse.
inverses_v2.jpg

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Tutor Answer

Homework_help
School: Duke University

Please find the attached documents and let me know if this is satisfactory.

Concepts:

Verifying inverse function by composition:
If 𝑓(𝑥 )𝑎𝑛𝑑 𝑔(𝑥) are two function such that
𝑓(𝑔 (𝑥 )) = 𝑔(𝑓(𝑥 )) = 𝑥
then, f(x) and g(x) are inverse to each other.
In other word, we can say: 𝑔(𝑥) = 𝑓 −1 (𝑥) 𝑎𝑛𝑑 𝑓(𝑥 ) = 𝑔−1 (𝑥)


−𝒙−𝟒

(C) 𝒇(𝒙) =

𝒙

−𝑥−4

Let 𝑦 =

𝑥

Interchanging variables x and y and solving for y.
=> 𝑥 =

−𝑦−4
𝑦

=> 𝑦𝑥 = −𝑦 − 4
=> 𝑦𝑥 + 𝑦 = −4
=> 𝑦(𝑥 + 1) = −4
=> 𝑦 = −

4
𝑥+1
𝟒

i.e. 𝒇−𝟏 (𝒙) = − 𝒙+𝟏
Check:
Let 𝑓(𝑥) =

−𝑥−4
𝑥

4

𝑎𝑛𝑑 𝑔(𝑥) = − 𝑥+1
4

=> 𝑓(𝑔(𝑥)) = 𝑓 (− 𝑥+1 ) =
And 𝑔(𝑓(𝑥 )) = 𝑔 (

−𝑥−4
𝑥

4
) −4
𝑥+1
4

𝑥+1

−(−

)=−

=

4
−𝑥−4
(
) +1
𝑥

4−4(𝑥+1)
𝑥+1
4

𝑥+1

=−

=

4
−𝑥−4+𝑥
𝑥

Since 𝑓(𝑔(...

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Review

Anonymous
Thanks for the help.

Anonymous
Outstanding. Studypool always delivers quality work.

Anonymous
Tutor was very helpful and took the time to explain concepts to me. Very responsive, managed to get replies within the hour.

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