Question

Select the correct choice and fill in the blank if necessary

Answer 1

Given

$f(x)=\frac{x+6}{x-7}$

Recall

The horizontal line test can be used to determine if a function is one-to-one given a graph. Simply superimpose a horizontal line onto a graph and see if it intersects the graph at more than one point. If it does, the graph is not one-to-one and if it only intersects at one point, it will be one-to-one.

The graph

It passed the horizontal line test, therefore is one to one function

Part B

$f(x)=\frac{x+6}{x-7}$

Step 1

Replace f(x) with y

$y=\frac{x+6}{x-7}$

Step 2

Inter change y and x

$x=\frac{y+6}{y-7}$

Step 3

Make y the subject

$\begin{gathered} x=\frac{y+6}{y-7} \\ x(y-7)=y+6 \\ xy-7x=y+6 \\ xy-y=6+7x \\ y(x-1)=6+7x \\ divide\text{ both sides by x-1} \\ y=\frac{6+7x}{x-1} \end{gathered}$

Step 4

Replace y with f^-1

$f^{-1}(x)=\frac{6+7x}{x-1}$

The final answer

$f^{-1}(x)=\frac{6+7x}{x-1}$