91Ó°ÊÓ

The given composite function is:

$$\begin{gathered} f\left(x\right)=x^2+4 \\ g\left(x\right)=\sqrt{x-2} \end{gathered}$$

The domain of $f$ is the set of all real numbers andg is $\left\{x\overline{)xâ‰\yen 1}\right\}$.

$(f∘g)\left(x\right)=f\left(g\right(x\left)\right)$

Now substitute$g\left(x\right)=\sqrt{x-2}$in the function$f\left(g\right(x\left)\right)$.

Then the function will become$f\left(\sqrt{x-2}\right)$.

Now replace x with $\sqrt{x-2}$in$f\left(x\right)=x^2+4$,

localid="1646238507463" $$\begin{gathered} f\left(\sqrt{x-2}\right)={(x-2)}^2+4 \\ =x-2+4 \\ =x+2 \end{gathered}$$

Therefore, the domain of $f∘g$is $\left\{x\overline{)xâ‰\yen 2}\right\}$and $(f∘g)\left(x\right)=x+2$.

$(g∘f)\left(x\right)=g\left(f\right(x\left)\right)$

Now substitute $f\left(x\right)=x^2+4$in the function $g\left(f\right(x\left)\right)$.

Then the function will become $g(x^2+4)$.

Now replace x with $(x^2+4)$in $g\left(x\right)=\sqrt{x-2}$,

$$\begin{gathered} g(x^2+4)=\sqrt{x^2+4-2} \\ =\sqrt{x^2+2} \end{gathered}$$

Therefore, the domain of $g∘f$is all real numbers and $(g∘f)\left(x\right)=\sqrt{x^2+2}$.

$(f∘f)\left(x\right)=f\left(f\right(x\left)\right)$

Now substitute $f\left(x\right)=x^2+4$in the function $f\left(f\right(x\left)\right)$.

Then the function will become $f(x^2+4)$.

Now replace x with $x^2+4$in $f\left(x\right)=x^2+4$,

localid="1646241500938" $$\begin{gathered} f(x^2+4)={(x^2+4)}^2+4 \\ =x^4+8x^2+20 \end{gathered}$$

Therefore the domain of localid="1646241724338" $f∘f$is all real numbers and $(f∘f)\left(x\right)=x^4+8x^2+20$.

$(g∘g)\left(x\right)=g\left(g\right(x\left)\right)$

Now substitute $g\left(x\right)=\sqrt{x-2}$in the function $g\left(g\right(x\left)\right)$.

Then the function will become $g\left(\sqrt{x-2}\right)$..

Now replace x with $\sqrt{x-2}$in $g\left(x\right)=\sqrt{x-2}$,

$g\left(\sqrt{x-2}\right)=\sqrt{\sqrt{x-2}-2}$

Solve the inequality to find x,

$$\begin{gathered} \sqrt{x-2}-2â‰\yen 0 \\ \sqrt{x-2}â‰\yen 2 \\ x-2â‰\yen 4 \\ xâ‰\yen 6 \end{gathered}$$

Therefore, the domain of$g∘g$is$\left\{x\overline{)xâ‰\yen 6}\right\}$and$(g∘g)\left(x\right)=\sqrt{\sqrt{x-2}-2}$.