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Composition of Functions
It is possible to combine two functions by adding, subtracting, multiplying or dividing two given functions. You can learn more about those operations by clicking here.

There is another way to combine two functions to create a new function. It is called composition of two functions. It is a process through which we will substitute an entire function into another function. An important skill to have in this lesson is evaluation of functions. You should click here if you need to review evaluation of functions.

First let’s get acquainted with the notation that is used for composition of functions. When we want to find the composition of two functions we use the notation . Another way to write this is . This is probably the more practical notation although the first notation is what appears most often in books. But to really understand what the composition of functions is, will work better.

Recall our notation for evaluating a function. If we are given a function and are asked to find , we would go to the f function, and everywhere there was an x, we would replace it with a 2.

So if our notation is now , that says go to the f function and everywhere there is an x, replace it with the function . It’s not a different process than used for evaluating functions it just becomes more awkward because of the notation.

Let's Practice:
  1. Given and find .
Solution: Remember that the notation is the same as . So we will go to our f function and replace the x with the g function . This means we now have
If we are then asked to evaluate at a particular point, we would simply plug in the value into our new function.
We should be very careful to note that order DOES matter when working with composition of functions. This is illustrated in the example below using the same two functions that we used in our first exercise.
  1. Given and find .
Solution: To find let’s use the notation . So in the g function, everywhere there is an x, substitute the f function.
This answer is definitely different that what was obtained in our first practice example. So the order in which two functions are composed is important.
  1. Given and find .
Solution: We know so we substitute and get
  1. Given and find .
Solution: We use and substitute to get
There are some special situations when and are the same. But it is still important to pay attention to the order of the composition. When we do get the same answer for  and and that answer is x, we have two functions that are inverses of each other. Inverses are discussed in another lesson. If you would like to learn more, click here.

For each problem below, find both: (a) and (b)

Example   and
What is your answer?
Example   and
What is your answer?
Example   and
What is your answer?
Example   and
What is your answer?
Example   and
What is your answer?

S Taylor

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