### Home > A2C > Chapter 11 > Lesson 11.3.1 > Problem11-132

11-132.

For the function$f ( x ) = \frac { \sqrt { x + 4 } } { 2 } - 1$, complete parts (a) through (d) below.

1. Sketch the graph and the inverse.

First, graph the function. Then, reflect the graph across
y = x.

2. Find the equation of the inverse function.

Replace f(x) with y, then switch the variables.

$x=\frac{\sqrt{y+4}}{2}-1$

Now solve for y.

Do not forget to replace y with $f^{ − 1}\left(x\right)$.

$f^{ −1}\left(x\right) = \left(2\left(x + 1\right)\right)^{2} − 4$

3. Determine the domain and range of the inverse.

See graph in part (a).

$x ≥ −1, y ≥ −4$

4. Compute $f ^{−1}\left(f \left(5\right)\right)$.

Substitute $5$ for $x$ in the function $f\left(x\right)$. Now substitute the answer you found for $f\left(x\right)$ for $x$ in the inverse function
f −1(x).

$f^{ −1}\left(f\left(5\right)\right) = 5$