Question

Find the inverse of the function. State any restrictions on the domain.

f(x) = x^2 + 9

Answer 1

The function is not invertible if you think of it as

$f:\mathbb{R}\to\mathbb{R}$

because it is not injective: $f(x)=f(-x)$

So, if we restrict the domain to the positive numbers only:

$f:\mathbb{R^+}\to\mathbb{R}$

Note that the range is actually $[9,\infty)$, so we can be more precise and write

$f:\mathbb{R^+}\to[9,\infty)$

To find the inverse, we solve the expression for x:

$y=x^2+9 \iff y-9=x^2 \iff x=\pm\sqrt{y-9}$

But since we restricted to positive values of x, the inverse is

$x=\sqrt{y-9}$