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Section 7.1 Composition

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

You already know about inputs and outputs of a function. Function composition uses 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.

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

When you compose functions together, the output from one function becomes the input for the other.

The notation \(f(g(x))\) is read as “\(f\) of \(g\) of \(x\)”. Here, we first evaluate \(g(x)\) to get an output, and then use that output as the input for the function \(f\text{.}\)

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

With function composition, the key is to remember to evaluate the inside function first. Evaluating a composite function 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 that output as the input for the function \(f\text{.}\)

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.

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

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