Question

The one-to-one functions g and h are defined as follows.g={(-9, - 4). (4. 1), (5, 8), (7, 5)}h(x)=2x-9Find the following.

Answer 1

Ok, so

We got the function g:

$g=\mleft\lbrace(-9,-4\mright),(4,1),(5,8),(7,5)\}$

First, let's find

$g^{-1}(5)$

The points we're going to analyze are the next one:

As you can see, the value of x which makes that g(x) equals to 5, is 7.

So,

$g^{-1}(5)=7$

Now, we have h(x):

$h(x)=2x-9$

To find the inverse, we solve that equation for x:

$\begin{gathered} y=2x-9 \\ y+9=2x \\ x=\frac{y+9}{2} \\ \\ h^{-1}(x)=\frac{x+9}{2} \end{gathered}$

So that's the inverse of h(x).

Finally, we have to find:

$(h^{-1}\circ h)(2)$

This is the same that if we write:

$h^{-1}(h(2))$

So we're going to evaluate the inverse function, in h(2).

We can find h(2) replacing:

$\begin{gathered} h(x)=2x-9 \\ h(2)=2(2)-9 \\ h(2)=-5 \end{gathered}$

Now, evaluate:

$h^{-1}(-5)$

This is:

$\begin{gathered} h^{-1}(x)=\frac{x+9}{2} \\ h^{-1}(-5)=\frac{-5+9}{2} \\ h^{-1}(-5)=\frac{4}{2}=2 \end{gathered}$

Therefore,

$(h^{-1}\circ h)(2)=2$