Question

find an equation for F minus one the inverse function. Verify that your equation is correct by showing that

Answer 1

Explanation

We are to first find the inverse of the function:

$f(x)=\frac{x+12}{x-4}$$\mathrm{A\:function\:g\:is\:the\:inverse\:of\:function\:f\:if\:for}\:y=f\left(x\right),\:\:x=g\left(y\right)\:$

To do so, we will follow the steps below:

Step 1:

$\begin{gathered} write\text{ the function interms of y} \\ y=\frac{x+12}{x-4} \end{gathered}$

Step2: Interchange x with y

$x=\frac{y+12}{y-4}$

Step 3: solve for y

$\begin{gathered} xy-4x=y+12 \\ xy-y=12+4x \\ y(x-1)=12+4x \\ y=\frac{12+4x}{x-1} \end{gathered}$

Thus, the inverse of the function is

$\begin{gathered} f^{-1}(x)^=\frac{12+4x}{x-1} \\ \\ for \\ x\ne1 \end{gathered}$

Part 2

$f(f^{-1}(x))=f(\frac{12+4x}{x-1})$

Simplifying further

$\begin{gathered} f(\frac{12+4x}{x-1})=\frac{\frac{12+4x}{x-1}+12}{\frac{12+4x}{x-1}-4}=x \\ Thus \\ f(\frac{12+4x}{x-1})=x \end{gathered}$

Also

$f^{-1}(f(x))=f^{-1}(\frac{x+12}{x-4})$

Simplifying further

$\begin{gathered} f^{-1}(\frac{x+12}{x-4})=\frac{12+4\times\frac{x+12}{x-4}}{\frac{x+12}{x-4}-1}=x \\ \\ Thus \\ f^{-1}(\frac{x+12}{x-4})=x \end{gathered}$