Find the domain of composite function

Question

$f(x) = \dfrac x {(x-3)}$; $g(x)= \dfrac {-7}{(x+7)}$

I found The domain of $(f \circ g)$ is $(- \infty, -28/3) \cup (-28/3 , -7) \cup (-7, \infty)$ which is correct.

Can someone please tell me the domain of $(g \circ f)(x)$ ? Show work if possible please.

Thanks!

Answer 1

For $g\circ f$ I will try to do this the short way...

First, the domain of $f$ is: $x\not=3$...

Second, the domain of $g$ is: $x\not={-7}$. Thus we cannot have $f(x)=-7$. So we solve for $x$.
Get $\frac x{x-3}=-7 \implies x=-7x+21 \implies 8x=21 \implies x=\frac{21}8$.

Conclusion: $x\not=\frac{21}8,3$

Answer 2

Domain of $(gof)(x)$ is $$ x\in (-\infty,21/8)\cup (21/8,3)\cup (3, \infty)$$