# Section7.1Composition¶ permalink

In this activity, we will explore the *composition of functions*.

You already know about *inputs* and *outputs* of a function. *Function composition* is a way to use the output of one function as the input for another function.

In the first exercise, you will use what you already know in order to use function composition in a story about an oil spill. ^{ 1 } .

Now continue the previous exercise in the next problem, where you will use function composition to make one function out of two.

##### Exercise7.1.2

In chapter 5, you studied transformations of a function. There, you took a function \(f(x)\) and shifted its graph left or right by adding a number to the input of the function. For instance, \(f(x – 3)\) represented shifting the graph \(y = f(x)\) to the right by \(3\) units.

However, this may also be thought of as the *composition* of two functions: \(f(x)\) and \(g(x) = x - 3\)

As a composition, this transformation occurs by substituting the function \(g(x) = x - 3\) into the function \(f(x)\text{:}\) \begin{equation*} f(x - 3) = f(g(x)) \end{equation*} So, we have already been using function composition, though we have waited until now to give it that name.

When you compose functions together, the output from one function becomes the input for the other. In the example above, we would first use a value of \(x\) in the *inside* function \(g(x) = x - 3\) to get an ouput. Then, we would take that output and use it as the input for the function \(f(x)\text{.}\)

When we write \(f(g(x))\text{,}\) we read it as “*\(f\) of \(g\) of \(x\)*”.

In the next exercise, you will see an animation of composing two functions.

##### Exercise7.1.3

Now, practice function *composition* in the next problem, remembering to work inside the parentheses first.

##### Exercise7.1.4

With *function composition*, the key is to remember to evaluate the *inside* function first. Evaluating functions is just like doing regular arithmetic — work inside the parentheses before doing anything else.

\(f(g(x))\) | First, evaluate \(g(x)\) to get an output, then evaluate \(f\) of that output. |

\(H(3x-12)\) | First, evaluate \(3x-12\) to get an output, then evaluate \(H\) of that output. |

\(f(N(w(x)))\) | First, evaluate \(w(x)\text{,}\) then use that output as the input for the function \(N\text{,}\) and then use |

In the next exercise, you will get practice evaluating a composite function from a table of values. Remember to evaluate a composite function from the *inside* to the *outside*.

##### Exercise7.1.5

In Exercise 7.1.1 and Exercise 7.1.2 of this activity, you used *composition* to find a numerical answer for the area of the oil spill *and* a formula which found the area as a function of \(t\text{.}\)

To find a numerical answer, you evaluated the *radius* function to get a number, and then used that number to evaluate the *area* function.

To find a formula, you just used the *radius* formula as the input for the *area* function. This gave a new formula, but not a particular numerical answer.

If we compose functions together, we think of the resulting formula as a new, single function, written in the form: \begin{equation*} W(J(x)) \end{equation*}

We call this new formula a *composite* function, because it is *composed* of two different functions.

in the next exercise, you will be composing two functions to make a new *composite* function. Pay careful attention to which function is being used as the input for the other function.

##### Exercise7.1.6

Next, use the graphs of two functions to evaluate different compositions.