*label*Mathematics

### 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.

### Unformatted Attachment Preview

## Tutor Answer

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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