Composite Functions

(How to find the expression of a composite function)


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: \[f\begin{bmatrix}g(x)\end{bmatrix} = \begin{pmatrix}f \circ g \end{pmatrix}(x)\]

How to build the expression for a composite function

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.


Tutorial: Constructing Composite Functions

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.

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.


Example

Given the functions \(f(x)\) and \(g(x)\) defined as: \[f(x) = 2x-7\]

and
\[g(x) = x^2-1\] Find an expression for:
  1. the composite function \(f\begin{bmatrix}g(x)\end{bmatrix}\).
  2. the composite function \(g\begin{bmatrix}f(x)\end{bmatrix}\).

Solution

  1. We find \(f\begin{bmatrix}g(x)\end{bmatrix}\) by replacing every \(x\), in the expression of \(f(x)\), by \(g(x)\) that's: \[\begin{aligned} f\begin{bmatrix}g(x)\end{bmatrix} & = 2g(x) - 7 \\ & = 2\begin{pmatrix}x^2 - 1 \end{pmatrix} - 7 \\ & = 2x^2 - 2 - 7 \\ f\begin{bmatrix}g(x)\end{bmatrix} & = 2x^2 - 9 \end{aligned}\]
  2. We find \(g\begin{bmatrix}f(x)\end{bmatrix}\) by replacing every \(x\), in the expression of \(g(x)\), by \(f(x)\) that's: \[\begin{aligned} g\begin{bmatrix}f(x)\end{bmatrix} & = \begin{pmatrix}f(x) \end{pmatrix}^2 - 1 \\ & = \begin{pmatrix} 2x- 7 \end{pmatrix}^2 - 1 \\ & = 4x^2 - 28x + 49 - 1 \\ g\begin{bmatrix}f(x)\end{bmatrix} & = 4x^2 - 28x +48 \end{aligned}\]


Exercise 1

  1. Given the functions \(f(x) = 3x+5\) and \(g(x)=x^2+4\), find an expression for:
    1. the composite function \(f\begin{bmatrix}g(x)\end{bmatrix}\)
    2. the composite function \(g\begin{bmatrix}f(x)\end{bmatrix}\)

  2. Given the functions \(f(x) = \sqrt{x}\) and \(g(x) = 3x - 9\), find an expression for:
    1. the composite function \(\begin{pmatrix}f \circ g \end{pmatrix}(x)\)
    2. the composite function \(\begin{pmatrix}g \circ f \end{pmatrix}(x)\)

  3. Given the functions \(f(x) = \frac{5}{x+2}\) and \(g(x)=x^2+1\), find an expression for:
    1. the composite function \(f\begin{bmatrix}g(x)\end{bmatrix}\)
    2. the composite function \(g\begin{bmatrix}f(x)\end{bmatrix}\)

  4. Given the functions \(f(x) = 3x+5\) and \(g(x)=-x^2+1\), find an expression for:
    1. the composite function \(f\begin{bmatrix}g(x)\end{bmatrix}\)
    2. the composite function \(g\begin{bmatrix}f(x)\end{bmatrix}\)

  5. Given the functions \(f(x) = \frac{3}{2x}\) and \(g(x) = 4x + 5\), find an expression for:
    1. the composite function \(\begin{pmatrix}f \circ g \end{pmatrix}(x)\)
    2. the composite function \(\begin{pmatrix}g \circ f \end{pmatrix}(x)\)

Answers Without Working

    1. For \(f\begin{bmatrix}g(x)\end{bmatrix}\), we find: \[f\begin{bmatrix}g(x)\end{bmatrix} = 3x^2+17\]
    2. For \(g\begin{bmatrix}f(x)\end{bmatrix}\), we find: \[g\begin{bmatrix}f(x)\end{bmatrix} = 9x^2+30x + 29\]

    1. For \(\begin{pmatrix}f \circ g \end{pmatrix}(x)\), we find: \[\begin{pmatrix}f \circ g \end{pmatrix}(x) = \sqrt{3x-9}\]
    2. For \(\begin{pmatrix}g \circ f \end{pmatrix}(x)\), we find: \[\begin{pmatrix}g \circ f \end{pmatrix}(x) = 3\sqrt{x}-9\]

    1. For \(f\begin{bmatrix}g(x)\end{bmatrix}\), we find: \[f\begin{bmatrix}g(x)\end{bmatrix} = \frac{5}{x^2+3}\]
    2. For \(g\begin{bmatrix}f(x)\end{bmatrix}\), we find: \[g\begin{bmatrix}f(x)\end{bmatrix} = \frac{x^2+4x+29}{x^2+4x+4}\]


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