Q.If $f(x) = 2x + 3$ and $g(x) = x^2 - 1$, find $(g \circ f)(x)$ and $(f \circ g)(x)$. Are they equal?

Given $f:A\to B$ and $g:B\to C$, the composite function $(g\circ f)(x)=g(f(x))$ applies $f$ first and then $g$. Composition is generally not commutative ($f\circ g\ne g\circ f$) but is associative. The domain of a composite can be smaller than the domain of either function alone, since the intermediate output of the inner function must also lie in the domain of the outer function. When a composite formula and one of the two functions are given, the other function is found by treating the composite as an equation and solving algebraically. …