A composition of functions is formed by taking the outputs of one function and converting them into the inputs of another function. These functions can be very useful when we have to model different processes with different functions.

Here, we will explore a brief overview of function composition and we will learn how to get a composition if we have two functions. In addition, we will look at several examples with answers in order to master the process used to obtain the composition of functions.

## Summary of composition of functions

The composition of functions is an operation where two functions like and generate a new function like in such a way that we have .

This means that the function is applied to the function . This means that basically a function is applied to the result of another function.

**Symbol:** A composition of functions is also denoted as , where the small circle, , is the symbol of the composition of functions. We cannot replace the circle with a point (·) since this indicates the product of two functions.

**Domain:** The composition is read as “*f* of *g* of *x*“. In this composition, the domain of the function *f* becomes since the domain is the set of all input values of the function.

To apply the composition , we carry out the following two steps:

**Step 1:** We apply the function *g* to the input *x* and obtain the result as the output.

** Step 2:** We apply the function

*f*using as the input and we get the result as the output.

## Composition of functions – Examples with answers

The following function composition examples can be used to fully understand the process used to obtain a function composition. It is advisable to try to solve the exercises yourself before looking at the answer.

**EXAMPLE 1**

Find the composition is we have the functions and .

##### Solution

To find the combination , we have to use the outputs of the function on the inputs of . Therefore, we use as the inputs of the function *f* and we have:

**EXAMPLE 2**

Find the composition is we have the functions and .

##### Solution

The composition can also be written as . Therefore, we have to take the output of and use it as the input of . We start by substituting each value of *x* in function *f* for the function *g*:

Now, we can apply the exponent to expand and simplify to the function:

**EXAMPLE 3**

We have the functions and . Find the composition .

##### Solution

Similar to the previous exercise, we know that the composition can be written as . Therefore, we replace each *x* in the function with the function .

Then, we expand the parentheses with the exponent and simplify the composition of functions:

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**EXAMPLE 4**

Find the composition of functions is we have and .

##### Solution

In this case, we have the composition , which is equal to . Therefore, we use the output of as the input of :

**EXAMPLE 5**

We have the functions and . Find the composition .

##### Solution

We have to take the function and use it as the input of . In this case, the function has a square root.

Let’s see what happens when we expand and simplify:

**EXAMPLE 6**

Calculate the composition if we have the functions and .

##### Solution

We know that this composition is equivalent to having . Therefore, we use the outputs of as the inputs of . This case is interesting since we have a square root that goes inside another square root:

We can write radicals as exponential expressions with fractional exponents and then apply the power of a power rule to simplify:

Now, we rewrite the exponential expression as the fourth root:

**EXAMPLE 7**

If we have , find the composition of functions .

##### Solution

We start by removing the parentheses using the distributive property:

Here we have several terms with the variable *x* with different powers. We have to make sure to combine only the terms that have the same power:

## Composition of functions – Practice problems

Put into practice what you have learned about the composition of functions by solving the following problems. Choose an answer and check it to see if you got the correct answer. Check out the solved examples above in case you need help.

## See also

Interested in learning more about functions? Take a look at these pages:

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