CPM Homework Help : INT3 Problem 8-96

What is the inverse of $g\left(x\right) = \left(x + 1\right)^{2} – 3$ for $x\ge-1$ ? Sketch graphs of $g\left(x\right)$ and its inverse on the same set of axes, and then state the domain and range of the inverse function.

Step 1:

Let $g\left(x\right) = y$ . Then interchange $x$ and $y$ .
$x = \left(y + 1\right)^{2} − 3$

Step 2:

Add $3$ to both sides of the equation.
$x + 3 = \left(y + 1\right)^{2}$

Step 3:

Take the square root of both sides.

$y + 1 = \sqrt{x + 3}$

$g^{-1}(x) = \sqrt{x + 3} - 1$

Answer:

D: $x\ge−3$
R: $y\ge−1$

2 increasing curves, First starts at the point (negative 3, comma negative 1), rises slowly, opening down. Second starts at the point (negative 1, comma negative 3), rising quickly, opening up.