The exercise wants us to prove that f(g(x)) = x and g(f(x)) = x, then let's prove it.
First, let's start with f(g(x)) = x. Let's do f(g(x)), we must find that it's equal to x.
[tex]\begin{gathered} f(x)=x^2-3 \\ g(x)=\sqrt{x+3} \\ \\ f(g(x))=(\sqrt{x+3})^2-3 \end{gathered}[/tex]
Now we simplify that expression, then
[tex]\begin{gathered} f(g(x))=(\sqrt{x+3})^2-3 \\ \\ f(g(x))=x+3-3 \\ \\ f(g(x))=x \end{gathered}[/tex]
Then we can confirm that f(g(x)) = x. Now let's do the same for g(f(x)), then
[tex]\begin{gathered} g(x)=\sqrt{x+3} \\ f(x)=x^2-3 \\ \\ g(f(x))=\sqrt{(x^2-3)+3} \\ \\ g(f(x))=\sqrt{x^2-3+3} \\ \\ g(f(x))=\sqrt{x^2} \\ \\ g(f(x))=x \end{gathered}[/tex]
As we expected, it's also true, hence
[tex]\begin{gathered} f(g(x))=x \\ g(f(x))=x \end{gathered}[/tex]
