Put simply, a composite function is a function of a function.
The idea is to place a function inside another function. To do this we replace every \(x\) we see inside a function by another function.
Given two functions \(f(x)\) and \(g(x)\) we can make two (usually different) composite functions: \[f\begin{bmatrix}g(x)\end{bmatrix} \quad \text{and}\quad g\begin{bmatrix}f(x)\end{bmatrix}\] The first of the two is read "\(f\) of \(g\) of \(x\)" and the second "\(g\) of \(f\) of \(x\)". We'll often see these written in the alternative notation: \[\begin{pmatrix}f \circ g \end{pmatrix}(x) \quad \text{and} \quad \begin{pmatrix}g \circ f \end{pmatrix}(x)\] Note: either of the two notations refer to the same thing, meaning: \[f\begin{bmatrix}g(x)\end{bmatrix} = \begin{pmatrix}f \circ g \end{pmatrix}(x)\]
Given two functions, \(f(x)\) and \(g(x)\), we construct the composite function: \[f\begin{bmatrix}g(x)\end{bmatrix}\] By replacing every \(x\), in the expression for \(f(x)\) by the entire function \(g(x)\).
Similarly, to construct: \[g\begin{bmatrix}f(x)\end{bmatrix}\] replace every \(x\), in the expression for \(g(x)\) by the entire function \(f(x)\).
The method we've just read is illustrated in the following tutorial.
In the following tutorial we learn what a composite function is as well as how to contruct a composite function, given two functions \(f(x)\) and \(g(x)\).
Although it was mentionned in the tutorial we've just seen, it's worth making a note of the following important result.
Given two functions \(f(x)\) and \(g(x)\) it is important to know that in general: \[f\begin{bmatrix}g(x) \end{bmatrix} \neq g\begin{bmatrix}f(x) \end{bmatrix} \] Note: it is possible for the two composite functions to be equal, but most of the time they won't be.
Given the functions \(f(x)\) and \(g(x)\) defined as: \[f(x) = 2x-7\]
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