The “composition of functions” is the process of combining two or more **functions** in a way where the output of one **function** is the input for the next **function**.

## Related Topics

- How to Add and Subtract Functions
- How to Multiply and Dividing Functions
- How to Solve Function Notation
- How to find Inverse of a Function

## Step by step guide to solve Composition of Functions

- The term “composition of functions” is simply the combination of two or more functions where the output from one function becomes the input for the next function.
- The notation used for composition is: \(\color{blue}{(f \ o \ g)(x)=f(g(x))}\)

### Composition of Functions – Example 1:

Using \(f(x)=x-2\) and \(g(x)=x\), find: \(f(g(2))\)

**Solution:**

\((f \ o \ g)(x)=f(g(x)) \)

Then: \((f \ o \ g)(x)=f(g(x))=f(x)=x-2\)

Substitute \(x\) with \(2: (f \ o \ g)(2)=2-2=0\)

### Composition of Functions – Example 2:

Using \(f(x)=x+8\) and \(g(x)=x-2\), find: \(g(f(4))\)

**Solution:**

\((f \ o \ g)(x)=f(g(x)) \)

Then: \((g \ o \ f)(x)=g(f(x))=g(x+8)\), now substitute \(x\) in \(f(x)\) by \(x+8\). Then: \(g(x+8)=(x+8)-2=x+8-2=x+6\)

Substitute \(x\) with \(4: (g \ o \ f)(4)=g(f(x))=4+6=10\)

### Composition of Functions – Example 3:

Using \(f(x)=x+2\) and \(g(x)=4x\), find: \(f(g(1))\)

**Solution:**

\((f \ o \ g)(x)=f(g(x)) \)

Then: \((f \ o \ g)(x)=f(g(x))=f(4x)=4x+2\)

Substitute \(x\) with \(1: (f \ o \ g)(1)=4+2=6\)

### Composition of Functions – Example 4:

Using \(f(x)=5x+4\) and \(g(x)=x-3\), find: \(g(f(3))\)

**Solution:**

\((f \ o \ g)(x)=f(g(x))\)

Then: \((f \ o \ g)(x)=g(f(x))=g(5x+4)\), now substitute \(x\) in \(g(x)\) by \(5x+4\). Then: \(g(5x+4)=(5x+4)-3=5x+4-3=5x+1\)

Substitute \(x\) with \(3: (g o f)(x)=g(f(x))=5x+1=5(3)+1=15=1=16\)

## Exercises for Solving Composition of Functions

### Using \(f(x) = 5x + 4\) and \(g(x) = x – 3\), find:

- \(\color{blue}{f(g(6))}\)
- \(\color{blue}{f(f(8))}\)
- \(\color{blue}{g(f(– 7))}\)
- \(\color{blue}{g(f(x))}\)

### Download Composition of Functions Worksheet

## Answers

- \(\color{blue}{19}\)
- \(\color{blue}{224}\)
- \(\color{blue}{-34}\)
- \(\color{blue}{5x+1}\)