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### Course: Algebra (all content) > Unit 7

Lesson 15: Composing functions (Algebra 2 level)# Intro to composing functions

Learn why we'd want to compose two functions together by looking at a farming example.

Cam is a farmer. Each year he plants seeds that turn into corn. The function below gives the amount of corn, $C$ , in kilograms (kg), that he expects to produce if he plants corn on $a$ acres of land.

For example, if Cam plants two, he expects to produce $C(2)=7500(2)-1500=13,500$ $\text{kg}$ of corn.

What Cam really wants to know is how much money he will make from selling this corn. So he uses the following function to predict the amount of money, $M$ , in dollars, that he will earn from selling $c$ kilograms of corn.

So if Cam produces $\mathrm{13,500}\text{kg}$ of corn, he can expect to make $M(\mathrm{13,500})=0.9(\mathrm{13,500})-50=\mathrm{\$}\mathrm{12,100}$ .

Notice that Cam has to use $C$ , takes acres to corn, while the second function, $M$ , takes corn to money.

*two*separate functions to get from acres planted to expected earnings. The first function,Wouldn't it be great if Cam could write a function that turned planted acres directly into expected earnings?

# Creating a new function

We can indeed find the function that takes acres planted directly to expected earnings! To find this new function, let's think about the most general question: how much money does Cam expect to make if he plants corn seed on $a$ acres of land?

Well, if Cam plants corn on $a$ acres, he expects to produce $C(a)$ kilograms of corn. And if he produces $C(a)$ kilograms of corn, he expects to make $M(C(a))$ dollars.

So, to find a general rule that converts $a$ acres directly into expected earnings, we can find the expression $M(C(a))$ .

But just how do we do this? Well, notice that in the expression $M({C(a)})$ , the input of function $M$ is ${C(a)}$ . So, to find this expression, we can substitute ${C(a)}$ in for ${c}$ in function $M$ .

So the function $M(C(a))=6750a-1400$ converts acres planted directly into expected earnings. Let's use this new function to predict the amount of money that Cam would make from planting corn on two acres.

Cam can expect to make $\mathrm{\$}\mathrm{12,100}$ from planting corn on two acres of land, which is consistent with our previous work!

# Defining composite functions

We just found what is called a

**composite function**. Instead of substituting acres planted into the corn function, and then substituting the amount of corn produced into the money function, we found a function that takes the acres planted directly to the expected earnings.We did this by substituting $C(a)$ into function $M$ , or by finding $M(C(a))$ . Let's call this new function $M\circ C$ , which is read as "$M$ composed with $C$ ".

We now know that $(M\circ C)(a)=M(C(a))$ . This, in fact, is the formal definition of function composition!

# Visualizing the two methods

Here's a visual to help interpret the above definition.

Using both functions $C$ and $M$ , function $C$ —the corn function—takes two to 13,500. Then, function $M$ —the money function—takes 13,500 to $\mathrm{\$}$ 12,100.

Using the composite function, we see that function $M\circ C$ takes two directly to $\mathrm{\$}$ 12,100.

The two are equivalent!

# Now let's practice some problems.

### Problem 2

Ben is a potato farmer. The function $P(a)=\mathrm{25,000}a-1000$ gives the amount of potatoes, $P$ , in kilograms, that he expects to produce from planting potatoes on $a$ acres of land. The function $M(p)=0.2p-200$ gives the amount of money, $M$ , in dollars, that Ben expects to make if he produces $p$ kilograms of potatoes.

### Problem 3

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