Question

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

Answer 1

Step-by-step explanation

We are to first find the inverse of the function:

`f(x)=(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=(x+12)/(x-4) \end{gathered}`

Step2: Interchange x with y

`x=(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=(12+4x)/(x-1) \end{gathered}`

Thus, the inverse of the function is

`\begin{gathered} f^(-1)(x)^=(12+4x)/(x-1) \\ \\ for \\ x\\e1 \end{gathered}`

Part 2

`f(f^(-1)(x))=f((12+4x)/(x-1))`

Simplifying further

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

Also

`f^(-1)(f(x))=f^(-1)((x+12)/(x-4))`

Simplifying further

`\begin{gathered} f^(-1)((x+12)/(x-4))=(12+4*(x+12)/(x-4))/((x+12)/(x-4)-1)=x \\ \\ Thus \\ f^(-1)((x+12)/(x-4))=x \end{gathered}`