CPM Homework Help : INT3 Problem 6-84

What is the equation for the inverse of the function $f(x)=2 \sqrt { \frac { ( x - 3 ) } { 4 } }+ 1$ ? Sketch the graph of both the original and the inverse functions.

Step 1:

Change $f(x)$ to $y$ . Then interchange the $x$ and $y$ .

$x=2\sqrt{\frac{\left(y-3\right)}{4}}+1$

Step 2:

Subtract $1$ from each side, and then square each side to get rid of the square root.

$\left(x-1\right)^2=\left(2\sqrt{\frac{y-3}{4}}\right)^2$

Step 3:

Cancel the $4$ in the numerator and denominator, then add $3$ to each side of the equation.

$\left(x-1\right)^2=4\cdot\frac{\left(y-3\right)}{4}$

Answer:

$f^{-1}\left(x\right)=\left(x-1\right)^2+3$

Are there any domain restrictions?