# Intro to combining functions

CCSS Math: HSF.BF.A.1b

Become familiar with the idea that we can add, subtract, multiply, or divide two functions together to make a new function.

Just like we can add, subtract, multiply, and divide numbers, we can also add, subtract, multiply, and divide functions.

# The sum of two functions

## Part 1: Creating a new function by adding two functions

Let's add ${f(x)=x+1}$ and ${g(x)=2x}$ together to make a new function.

Let's call this new function $h$. So we have:

## Part 2: Evaluating a combined function

We can also evaluate combined functions for particular inputs.
Let's evaluate function $h$ above for $x=2$. Below are two ways of doing this.

**Method 1:**Substitute $x=2$ into the combined function $h$.

**Method 2**: Find $f(2)$ and $g(2)$ and add the results.

Since $h(x)=f(x)+g(x)$, we can also find $h(2)$ by finding $f(2) +g(2)$.

First, let's find $f(2)$:

Now, let's find $g(2)$:

So $f(2)+g(2)=3+4=\greenD7$.

Notice that substituting $x =2$ directly into function $h$ and finding $f(2) + g(2)$ gave us the same answer!

# Now let's try some practice problems.

In problems 1 and 2, let $f(x)=3x+2$ and $g(x)=x-3$.

#### Problem 1

#### Problem 2

# A graphical connection

We can also understand what it means to add two functions by looking at graphs of the functions.

The graphs of $y=m(x)$ and $y=n(x)$ are shown below. In the first graph, notice that $m(4)=2$. In the second graph, notice that $n(4)=5$.

Let $p(x)=m(x)+n(x)$. Now look at the graph of $y=p(x)$. Notice that $p(4)=\blueD 2+\maroonD 5=\purpleD7$.

Challenge yourself to see that $p(x) = m(x) + n(x)$ for every value of $x$ by looking at the three graphs.

## Let's practice.

#### Problem 3

The graphs of $y=f(x)$ and $y=g(x)$ are shown below.

# Other ways to combine functions

All of the examples we've looked at so far create a new function by adding two functions, but you can also subtract, multiply, and divide two functions to make new functions!

For example, if $f(x)=x+3$ and $g(x)=x-2$, then we can not only find the sum, but also ...

**... the difference.**

**... the product.**

**... the quotient**.

In doing so, we have just created three new functions!