Question

Are g(x) and f(x) inverses of each other? Show all work always and explain in

words as you go. *
g(x)=4-3/2x
f(x)=1/2x+3/2

Answer 1

No,
`g(x)` and
`f(x)` are not inverses of each other.

Explanation:

Given functions:

`g(x)=4-(3)/(2)x`

`f(x)=(1)/(2)x+(3)/(2)`

To tell whether
`g(x)` and
`f(x)` are inverse of each other.

Solution:

If
`g(x)` and
`f(x)` are inverse of each other, then:

`f(x)=g^(-1)x` or
`g(x)=f^(-1)(x)`

Thus, to check if they are inverse of each other, we will find inverse of function
`g(x)` and see if it is =
`f(x)`

We have:

`g(x)=4-(3)/(2)x`

In order to find inverse we will replace
`g(x)` with
`y`

`y=4-(3)/(2)x`

Then we switch
`x` and
`y`

`x=4-(3)/(2)y`

Then, we solve for
`y`

Subtracting both sides by 4.

`x-4=4-4-(3)/(2)y`

`x-4=-(3)/(2)y`

Multiplying both sides by 2.

`2(x-4)=2*-(3)/(2)y`

Using distribution:

`2x-8=-3y`

Dividing both sides by -3.

`(2x)/(-3)-(8)/(-3)=(-3y)/(-3)`

`-(2)/(3)x+(8)/(3)=y`

Thus inverse of function
`g(x)` can be given as:

`g^(-1)(x)=(8)/(3)-(2)/(3)x`

Since
`g^(-1)(x)\\eq f(x)`, hence they are not inverse functions of each other.