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Composite Functions

Composite Functions

Given two functions, $f(x)$ and $g(x)$, It is possible to define the composite function $f\circ g(x)$ as follows $$ f\circ g(x) := f(g(x)) $$ Note that for this to be valid, the range of $g(x)$ must be a subset of the domain of $f(x)$ We also observe that in general, $$ f\circ g(x) \neq g\circ f(x). $$ There are however notable exceptions to this, the most obvious example being $$ f\circ f^{-1}(x) = f^{-1}\circ f(x) = x $$

Composite Functions 1

Determine $f\circ g(x)$, stating its range and domain, given that $f(x) = x^2 - 4,\;x\in \mathbb{R}$ and $g(x) = \frac{1}{x-3},\;x\neq 3$
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We find the composite function as follows. $$ f\circ g(x) = f(g(x)) = f\left(\frac{1}{x-3}\right) = \left(\frac{1}{x-3}\right)^2 - 4 $$ We note that the domain of $g(x)$ is $x\in \mathbb{R}, x\neq 3$. The domain of $f(x)$ is $x\in \mathbb{R}$ so the domain of $f\circ g(x)$ is $x\in \mathbb{R}, x\neq 3$. The range of the function $f\circ g(x)$ is $$ f\circ g(x) > -4,\;f\circ g(x) \in \mathbb{R} $$