• Rewrite composite functions into its simpler parts

In the Integration using Substitution topic, we will learn all about using substitution as an integration method. Substitution is basically the process used to find the antiderivative of a function that was differentiated using the chain rule. That being said, it is important to be able to look at a composite function and identify the inside function and outside function. Usually, the inside function is what we set our substitution variable equal to.

Rewriting Composite Functions into Simpler Components

Understanding the structure of composite functions is essential for dissecting complex mathematical expressions into more manageable parts. A composite function is formed when one function is applied to the result of another function. Decomposing these functions helps in understanding and simplifying their operations.

Consider [latex]f\left(x\right)=\sqrt{5-{x}^{2}}[/latex]. This can be seen as a composition of two simpler functions:

• [latex]g\left(x\right)=5-{x}^{2}[/latex]
• [latex]h\left(x\right)=\sqrt{x}[/latex]

Here, [latex]f(x) = h(g(x))[/latex], where [latex]g(x)[/latex] is first evaluated, and then [latex]h(x)[/latex] is applied to the result. 

composite functions

A composite function is formed when the output of one function becomes the input of another. They are expressed as 

[latex]f(x) = h(g(x))[/latex],

with [latex]g(x)[/latex] being evaluated first and [latex]h(x)[/latex] applied to its output.

How To: Decompose Composite Functions

1. Identify the Outer Function: Determine the last operation applied in the function. This is your outer function [latex]h(x)[/latex]
2. Identify the Inner Function: Look for the operation inside the outer function. This operation, which is applied first, is your inner function [latex]g(x)[/latex]
3. Express as a Composition: Write the original function as [latex]f(x) = h(g(x))[/latex], where [latex]g(x)[/latex] is evaluated first, and its result is used as the input for [latex]h(x)[/latex]

Write [latex]f\left(x\right)=e^{4x-3}[/latex] as the composition of two functions.

Show Solution

We are looking for two functions, [latex]g[/latex] and [latex]h[/latex], so [latex]f\left(x\right)=h\left(g\left(x\right)\right)[/latex]. To do this, we look for a function inside a function in the formula for [latex]f\left(x\right)[/latex]. As one possibility, we might notice that the expression [latex]4x-3[/latex] is within the exponent of the exponential function. We could then decompose the function as

[latex]g\left(x\right)=4x-3\hspace{2mm}\text{and}\hspace{2mm}h\left(x\right)=e^{x}[/latex]

We can check our answer by recomposing the functions.

[latex]h\left(g\left(x\right)\right)=h(4x-3)=e^{4x-3}[/latex]

Write [latex]f\left(x\right)=\dfrac{4}{3-\sqrt{4+{x}^{2}}}[/latex] as the composition of two functions.

Show Solution

There are many possible answers, one potential answer is:

[latex]g\left(x\right)=\sqrt{4+{x}^{2}}[/latex]

[latex]h\left(x\right)=\dfrac{4}{3-x}[/latex]