We explore how when can combine two functions by composing them. That is, how we can plug one function into another to create a new function.

Introduction

Recall that a function, by definition, is a process that takes a collection of inputs and produces a corresponding collection of outputs in such a way that the process produces one and only one output value for any single input value. Because every function is a process, it makes sense to think that it may be possible to take two function processes and do one of the processes first, and then apply the second process to the result.

When we have a situation such as in the example above where we use the output of one function as the input of another, we often say that we have composed two functions. In addition, we use the notation to denote that a new function, , results from composing the two functions and .

a.
Let and . Determine a formula for that depends only on and not on or . What is the biggest difference between your work in this problem compared to the example above?
b.
Let and recall that . Determine a formula for that depends only on .
c.
Suppose that . Determine formulas for two related functions, and , so that .

Composing Two Functions

Whenever we have two functions, and , where the outputs of match inputs of , it is possible to link the two processes together to create a new process that we call the composition of and .

We sometimes call the “inner function” and the “outer function”. It is important to note that the inner function is actually the first function that gets applied to a given input, and then the outer function is applied to the output of the inner function. In addition, in order for a composite function to make sense, we need to ensure that the outputs of the inner function are values that it makes sense to put into the outer function so that the resulting composite function is defined.

In addition to the possibility that functions are given by formulas, functions can be given by tables or graphs. We can think about composite functions in these settings as well, and the following activities prompt us to consider functions given in this way.

Let functions and be given by the graphs below (which are each piecewise linear - that is, parts that look like straight lines are straight lines) and let and be given by the table below.

Compute each of the following quantities or explain why they are not defined.

a.
b.
c.
d.
e.
f.
g.
For what value(s) of is ?
h.
For what value(s) of is ?

Composing functions in content

In the late 1800s, the physicist Amos Dolbear was listening to crickets chirp and noticed a pattern: how frequently the crickets chirped seemed to be connected to the outside temperature. If we let represent the temperature in degrees Fahrenheit and the number of chirps per minute, we can summarize Dolbear’s observations with the following function, . Scientists who made many additional cricket chirp observations following Dolbear’s initial counts found that this formauls holds with remarkable accuracy for the snowy tree cricket in temperatures ranging from to . This function is called Dolbear’s Law.

In what follows, we replace with to emphasize that temperature is measured in Fahrenheit degrees.

The Celsius and Fahrenheit temperature scales are connected by a linear function. Indeed, the function that converts Fahrenheit to Celsius is

For instance, a Fahrenheit temperature of degrees corresponds to degrees Celsius.
Let be Dolbear’s function that converts an input of number of chirps per minute to degrees Fahrenheit, and let be the function that converts an input of degrees Fahrenheit to an output of degrees Celsius.
a.
Determine a formula for the new function that depends only on the variable .
b.
What is the meaning of the function you found in (a)?
c.
Let . How does a plot of the function compare to the that of Dolbear’s function? Sketch a plot of on the blank axes to the right of the plot of Dolbear’s function, and discuss the similarities and differences between them. Be sure to label the vertical scale on your axes.

Function Composition and Average Rate of Change

Recall that the average rate of change of a function on the interval is given by

In the graph below, we see the familiar representation of as the slope of the line joining the points and on the graph of .

In the study of calculus, we progress from the average rate of change on an interval to the instantaneous rate of change of a function at a single value; the core idea that allows us to move from an average rate to an instantaneous one is letting the interval shrink in size.

To think about the interval shrinking while stays fixed, we often change our perspective and think of as , where measures the horizontal difference from to .

This allows us to eventually think about getting closer and closer to , and in that context we consider the equivalent expression

for the average rate of change of on .
Let and .
a.
Compute and expand and simplify the result as much as possible by combining like terms.
b.
Determine the most simplified expression you can for the average rate of change of on the interval . That is, determine for and simplify the result as much as possible.
c.
Compute . Is there any valid algebra you can do to write more simply?
d.
Determine the most simplified expression you can for the average rate of change of on the interval . That is, determine for and simplify the result.