Question

1. is the inverse a function?

$g(x) = x^2 + 6$
yes
no

1. write the inverse function in the form $a \pm b\sqrt{cx + d}$, where $a$, $b$, $c$, and $d$ are constants.

simplify any fractions.
$g^{-1}(x) = $

Explanation:

Step1: Test inverse function validity

A relation is a function if each input has only one output. For $g(x)=x^2+6$, it fails the horizontal line test (e.g., $g(2)=10$ and $g(-2)=10$), so its inverse will have one input mapping to two outputs, meaning it is not a function.

Step2: Find inverse relation

Start with $y = x^2 + 6$. Swap $x$ and $y$:
$x = y^2 + 6$
Solve for $y$:
$y^2 = x - 6$
$y = \pm\sqrt{x - 6}$

Answer:

1. no
2. $\pm\sqrt{x - 6}$