Answer
$f(g(x))=\frac{3}{x^2-1}$; Domain: All real numbers $x$ such that $x \ne -1, 1$
$g(f(x))=\frac{9}{x^2}-1$; Domain: All real numbers $x$ such that $x \ne 0$
The two composite functions are not equal.
Work Step by Step
$f(g(x))$
$f(x^2-1)$
$\frac{3}{x^2-1}$
The denominator must not equal 0 so:
$x^2-1\ne0$
$x^2\ne1$
$x \ne -1, 1$
The domain is all real numbers $x$ such that $x \ne -1, 1$
$g(f(x))$
$g(\frac{3}{x})$
$(\frac{3}{x})^2-1$
$\frac{9}{x^2}-1$
The domain is all real numbers $x$ such that $x\ne0$ since the denominator must not equal 0
The two composite functions are not equal since they do not have the same output or domain.
