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=(y+9)/(2) \\ \\ h^(-1)(x)=(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)=(x+9)/(2) \\ h^(-1)(-5)=(-5+9)/(2) \\ h^(-1)(-5)=(4)/(2)=2 \end{gathered}`

Therefore,

`(h^(-1)\circ h)(2)=2`