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Composite and Inverse Functions

Neil Trivedi

Teacher

Neil Trivedi

Contents

Composite Functions

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Inverse Function

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Practice Question

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Composite Functions

Composite functions involve applying successive functions to an input. For example,  means , for which is applied first and then . This example is illustrated by the diagram below.

A flow diagram shows an input 𝑥 passed through function 𝑓and then function 𝑔 to produce the output 𝑔(𝑓(𝑥)), illustrating function composition.

Example 1:

The functions , , and  are defined by

Find:

a)  

Single Step:  means , so we apply first, and then .


b)  

Single Step:  means , so we apply first, and then .


c)  

Single Step:  means , so we apply  first, then , and then .


d)  

Single Step:  means , so we apply  first, and then apply again.

No answer provided.

Inverse Function

An inverse function, denoted as , performs the opposite of what the original function does. If  maps an input to an output , then the inverse function , maps  back to An example is shown below.

A mapping diagram shows 𝑓 sending 𝑥=5 to 𝑦=8 and the inverse function 𝑓−1 sending 8 back to 5, illustrating how inverse functions reverse mappings.
To find the inverse function , swap  and  in the equation for , then rearrange the equation to make the subject.

To note, an inverse can only exist for a one-to-one function.

Also, the graph of the inverse function is a reflection of the original function in the line . For example, below, we can see the graph of (represented by the red curve.) We can work out the inverse function to be (represented by the blue curve.) We can see that has been reflected in the line (shown dotted) to give .

A coordinate grid shows the exponential curve 𝑓(𝑥)=𝑒𝑥 and its inverse 𝑓−1(𝑥)=ln𝑥 reflected in the line 𝑦=𝑥, illustrating the relationship between inverse functions.

Example 2: 

Given

a) Find .

Step 1: Here, we have . Swap  and .

Step 2: Rearrange the equation to make  the subject.

Multiply both sides by .

Add to both sides.

Apply the natural logarithm  to both sides.

Therefore,

b) Solve .

Single Step: Use the logarithm power rule, , to rewrite the right-hand side, then exponentiate both sides using base to eliminate the natural logarithm and solve for .

No answer provided.

The domain of  is equal to the range of , and the domain of  is equal to the range of . This is because we are switching the inputs and outputs.

A table compares a function 𝑓 and its inverse 𝑓−1, showing that the domain of 𝑓 becomes the range of 𝑓−1 and the range of 𝑓 becomes the domain of 𝑓−1.

Example 3:

Given

a) Find an expression for .

Step 1: Here, we have . Swap and .

Step 2: Rearrange the equation to make the subject.

Therefore, .

b) Determine the domain and range of .

Step 1: Sketch the graph of .

A graph of 𝑓(𝑥) is shown starting at 𝑥=−9 and increasing gradually as 𝑥 increases, illustrating a function with a restricted domain.

Step 2: Identify the domain and range of .

The range of  is equal to the domain of . Thus, the range is .

To find the domain of , we need to find the range of . The domain of is restricted to , which means we will only consider the region of the graph where
. We need to find the values of when and .

When :

When :

Here is an annotated graph to illustrate this information. To note, the red region of the graph is the part we’re concerned about. Note that is included in the restricted domain, whereas isn't, so will be included in the range, but not .

A graph of 𝑓(𝑥) highlights the domain 0≤𝑥<7 and uses dashed guide lines to show that the corresponding range is from 𝑦=3 up to but not including 𝑦=4.

The range of , considering the restriction in the domain, is . Hence, the domain of is .

No answer provided.

Practice Question

Further Practice Questions

Domain and Range Introduction to Modulus Functions and Graphs