$(23)$ If the function $f$ has the rule $f(x) = \sqrt{x^2-9}$ and the function $g$ has the rule $g(x) = x+5$.
$a. $ find integers $c$ and $d$ such that $f(g(x)) =\sqrt{(x+c)(x+d)}$
$b.$ state the maximal domain for which $f(g(x)) $ is defined
For question $(23) \;b.$, in the above , I am struggling with finding the domain. The answer states that $x\le8$ and $x\ge2$.
However, for $f(x)$, you can’t have a negative square root over the real field so $x\le -3$ and $x\gt 3$. So because x has already got this restricted domain, I had gotten the answer that $f(g(x))$ would mean $x\le$ and $x\gt3$...
I know this is a very basic question but I just need some clarification.
Thanks!
I think you must have made a mistake at ($a$) or haven't noticed how $f$ and $f\circ g$ may have different definition domains. You noticed how $f$ is defined except for $-3 \leq x\leq 3$. Then $f\circ g$ is defined for $-3\leq g(x) \leq 3$ which is $-8\leq x \leq -2$.
You may find that $f(g(x))= \sqrt{(x+2)(x+8)}$.
Then, as you stated we must have $(x+2)(x+8)\geq 0$ for $f\circ g$ to be defined. This means that $(x+2)$ and $(x+8)$ have the same sign which happens for $x\leq -8$ or $x\geq -2$ The maximal definition domain is $(-\infty,-8)\cup(-2,\infty)$.
The $-$ signs may come from an error in your question (for instance is it $g(x)=x+5$ or $g(x)=x-5$?).