March 31, 2021

Chapters

## What are Composite Functions?

In order to understand what a composite function is, let’s take a look at the most **basic function.**

This is a linear function. The table below defines the **elements** inside of a linear function.

A | - | y | output |

B | 3 | m | slope |

C | - | x | input |

D | 1 | b | y-intercept |

A function takes numbers as inputs, performs a **transformation** on them, and results a number as output. Let’s plug in some numbers as inputs to our linear function to test it out.

x | y |

-3 | -8 |

-2 | -5 |

-1 | -2 |

0 | 1 |

1 | 4 |

2 | 7 |

3 | 10 |

Now, let’s look at a composite function. A composite function is **applying** one function to another. That is, we get the output of one function and plug it into another function.

Step 1 | Plug in a number to f(x) |

Step 2 | Get the result of f(x) |

Step 3 | Plug the result into g(x) |

Let’s take a look at an example using our **linear function** above. Take the following two functions.

Function 1 | f(x) = 3x + 1 |

Function 2 | g(x) = 5x |

As an example, let’s solve **g(f(x))** with the input 2.

1 | f(2) = 3(2) + 1 = 7 |

2 | g(7) = 5(7) |

3 | = 35 |

While this was pretty easy, solving composite functions can get more complex with more complex functions. This is why, instead of the **three steps** we performed above, we can just simplify the formula first.

1 | g(f(x) = 5(f(x)) |

2 | 5(f(x)) = 5(3x+1) |

3 | = 15x + 5 |

**Plugging** in the number 2, we can see that we get the same result:

The **notation** for composite functions is written in the table below.

Take the result of f(x) | f(x) = 3x g(x) = x + 5 g(f(2)) = 3(2) + 5 = 11 | |

The same as above | f(x) = x + 2 g(x) = 4x g(f(1)) = 4(1 + 2) = 12 |

## Common Composite Functions

Now that you’ve been exposed to more functions, let’s go through a few common forms that composite functions can take on using **three functions** as an example.

f(x) | g(x) | h(x) |

Addition | Division | Addition, division, multiplication |

The first **example** is:

The process to solve this **composite** function is in the table below.

f(x) | g(f(2)) | |

2 + 3 = 5 | g(5) = = 0.2 |

The **second** example is:

The process to solve this composite function is in the table below.

g(x) | f(g(2)) | |

f() = = 3.2 |

The **third example** is:

The process to solve this composite function is in the table below.

f(x) | g(f(1)) | h(g(f(-2))) | |

-2+3 = 1 | g(1) = = 1 | h(1) = = 2 |

The **last example** is:

The **process** to solve this composite function is in the table below.

Inverse | Process | Comparison | |

This is the inverse of a function, which means we need to take the inverse | f(x) = x + 3 y = x + 3 y - 3 = x | To get the inverse, we simply need to get x on one side. | f(2) = 2 + 3 = 5 = 5 - 3 = 2 |

## Problem 1

Now that you know more about composite functions, write your own function for f(x). Next, calculate the **composite** function using **g(x)** that is given:

Plug in a few numbers and see what you come up with.

## Problem 2

Solve the following function for the following values for x, making sure to show your work.

Values for x:

- 4
- 1
- 2

## Problem 3

You are given the following functions:

f(x) | g(x) | |

Function | x + 2 | x |

Simplify the composite function:

## Problem 4

You are given the following functions:

f(x) | g(x) | h(x) | |

Function |

Solve the composite function:

## Problem 5

You have the following function:

Calculate the **inverse** of this function.

## Solution Problem 1

You were asked to come up with your own function. Then, calculate **g(f(x))** for a couple of values.

Let’s take the following function as an example:

Here are some sample values.

f(x) | g(f(x)) | |

1 | f(1) = 1 + 2 = 3 | g(3) = 4 |

4 | f(3) = 3 + 2 = 5 | g(5) = 3.25 |

10 | f(6) = 6 + 2 = 8 | g(8) = 3.1 |

## Solution Problem 2

You were asked to show the following values of x for the following function:

Values for x:

- f(4) = = 1.3
- f(1) = = 0.7
- f(2) = = 0.9

Here is the plot for this function:

## Solution Problem 3

You needed to **simplify** the composite function for these two functions:

f(x) | g(x) | |

Function | x + 2 | x |

## Solution Problem 4

You had to solve the composite function given these **three functions:**

f(x) | g(x) | h(x) | |

Function |

Solve the composite function:

## Solution Problem 5

Let’s take the inverse of the following function: