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.

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.

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

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.

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.

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.