Q.Let $\mathbb{R}$ be the set of real numbers and $f : \mathbb{R} \to \mathbb{R}$, $g : \mathbb{R} \to \mathbb{R}$ are two mappings such that $f(x) = |x| - x^2$, $g(x) = 2x + 3;\; \forall x \in \mathbb{R}$, then the value of $(g \circ f)(-3)$ is

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. …