 # 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. inverses_v2.jpg

### Unformatted Attachment Preview

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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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. Studypool 4.7 Trustpilot 4.5 Sitejabber 4.4 Brown University

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