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, watch it now.


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.

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: \[g\begin{bmatrix}f(x)\end{bmatrix} = \frac{x^2+4x+29}{x^2+4x+4}\]