Question

I need to verify the identity functions and find the inverse of an equation

Answer 1

You have the following function:

`f(x)=2x-8`

In order to determine f^-1, replace x by y and f(x) by x and solve for y, as follow:

`\begin{gathered} x=2y-8 \\ x+8=2y \\ (x+8)/(2)=y \end{gathered}`

As you can notice, the obtained expression for y is the inverse function.

Hence:

f^-1(x) = (x + 8)/2

In order to verify the given identities, proceed as follow:

`\begin{gathered} (f\circ f^(-1))(x)=f(f^(-1)(x)) \\ =f((x+8)/(2)) \\ =2((x+8)/(2))-8 \\ =x+8-8 \\ =x \end{gathered}`

Now, for the other identity:

`\begin{gathered} (f^(-1)\circ f)(x)=f^(-1)(f(x)) \\ =f^(-1)(2x-8) \\ =(2x-8+8)/(2) \\ =(2x)/(2) \\ =x \end{gathered}`