Find the composite functions f @ g and g @ f. Find the domain of each composite function. Are the two composite functions equal? f(x) = x^2 − 1 g(x) = −x

Find the composite functions $f\circ g$ and $g\circ f$. Find the domain of each composite function. Are the two composite functions equal?
$f\left(x\right)={x}^{2}-1$
g(x) = −x
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To find the composite function, $f\circ g$ and $g\circ f$ and also their respective domains.
Solution:
The functions are $f\left(x\right)={x}^{2}-1$ and $g\left(x\right)=-x.$
The function $f\circ g$ can be evaluated as,
$f\circ g=f\left(g\left(x\right)\right)$
$=f\left(-x\right)$
$={\left(-x\right)}^{2}-1$
$={x}^{2}-1$
The function $g\circ f$ can be evaluated as,
$g\circ f=g\left(f\left(x\right)\right)$
$=g\left({x}^{2}-1\right)$
$=-\left({x}^{2}-1\right)$
$=-{x}^{2}+1$
The obtained composite function are not equal. And the domain for both the function is complete real numbers as it is defined for each value of x.