How to find the Domain for this function

Question

my working out:

• x+a>0

• x>-a

• x^2-a^2 > 0

• x>|a|

• a^2-x^2>0

• |a|>x

hmm.. the answer is b ... how, why?

Answer 1

It's a rational function with square roots, so there are two important things to consider.

First off, the denominator cannot equal $0$. Secondly, the value under the radical must be non-negative.

So looking at the denominator, you know that $x\neq -a$.

Looking at the square roots, you need $x^2-a^2\geq 0$ and $a^2-x^2\geq 0\\\implies a^2=x^2.$

Now taking into account that $x\neq -a$, solve $a^2=x^2$.

$x^2=a^2\implies x=\pm a$,

but because of the denominator, we said $x\neq -a$.

Hence $D=\{a\}$.

Answer 2

Hint: Look at the top. You need $$ x^2 - a^2\ge 0\\ x^2 - a^2\le 0, $$hence $x^2 = a^2$.

Now look at the bottom: you need $x+a\neq 0$.