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# Composition of functions

The composition of functions is the successive application of functions. The resulting function is called a composite function.

The composite of $f(x)=2x$ and $g(x)=x^2$ is $h(x)=f(g(x))=2x^2$.

More formally, if $g:D_g\to R_g$ and $f:D_f\to R_f$ are two functions with $$R_g\subset D_f,$$ the composite function $f\circ g$ is the function from $D_g$ to $R_f$ defined by $$f\circ g (x) = f\big(g(x)\big).$$

The composite function is written either as $f\circ g$ or $fg$ and is read $f$ of $g$. $fg$ is often used for the product of functions, so $f\circ g$ should be preferred.

$h(x) = \sin(x^2)$ is the composite $\sin\circ g$ where $g(x) = x^2$. $\vert x^3\vert$ is the composite function $f\circ g$ with $f(x) =\vert x\vert$ and $g(x) = x^3$.