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

Answer:

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

Step-by-step explanation:

Given functions:

$g(x)=4-\frac{3}{2}x$

$f(x)=\frac{1}{2}x+\frac{3}{2}$

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:

$g(x)=4-\frac{3}{2}x$

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

$y=4-\frac{3}{2}x$

Then we switch $x$ and $y$

$x=4-\frac{3}{2}y$

Then, we solve for $y$

Subtracting both sides by 4.

$x-4=4-4-\frac{3}{2}y$

$x-4=-\frac{3}{2}y$

Multiplying both sides by 2.

$2(x-4)=2\times-\frac{3}{2}y$

Using distribution:

$2x-8=-3y$

Dividing both sides by -3.

$\frac{2x}{-3}-\frac{8}{-3}=\frac{-3y}{-3}$

$-\frac{2}{3}x+\frac{8}{3}=y$

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

$g^{-1}(x)=\frac{8}{3}-\frac{2}{3}x$

Since $g^{-1}(x)\neq f(x)$, hence they are not inverse functions of each other.