Answer:
[-5, 4) ∪ (4, ∞)
Step-by-step explanation:
Given functions:
[tex]f(x)=\dfrac{1}{x-3}[/tex]
[tex]g(x)=\sqrt{x+5}[/tex]
Composite function:
[tex]\begin{aligned}(f\:o\:g)(x)&=f[g(x)]\\ & =\dfrac{1}{\sqrt{x+5}-3} \end{aligned}[/tex]
Domain: input values (x-values)
For [tex](f\:o\:g)(x)[/tex] to be defined:
[tex]x+5\geq 0 \implies x\geq -5[/tex]
[tex]\sqrt{x+5}\neq 3 \implies x\neq 4[/tex]
Therefore, [tex]-5\leq x < 4[/tex] and [tex]x > 4[/tex]
⇒ [-5, 4) ∪ (4, ∞)
