1.9 Combining Functions

Introduction

Arithmetic with functions

Combining functions in context

Consider the functions $f$ and $g$ defined by the following table and the piecewise linear functions $p$ and $q$ defined by the following figure. Assume that the lines in the figure pass through whole number coordinates where they appear to do so; for example, $(2,2)$ lies on the graph of $q$, and $(3,-3)$ lies on the graph of $p$.

$x$ | 0 | 1 | 2 | 3 | 4

$f(x)$ | 5 | 10 | 15 | 20 | 25

$g(x)$ | 9 | 5 | 3 | 2 | 3

Let $h(x) = f(x) + g(x)$. Determine $h(3)$.

Determine the exact value of $(f+g)(0)$.

Determine the exact value of $(g-f)(1)$.

Determine the exact value of $(f \cdot g)(-1)$.

Are there any values of $x$ for which $\left( \frac{f}{g} \right)(x)$ is undefined? If not, explain why. If so, determine the values and justify your answer.

For what values of $x$ is $(f \cdot g)(x) = 0$? Why?

Are there any values of $x$ for which $(f-g)(x) = 0$? Why or why not?

Say that an investor is regularly purchasing stock in a particular company.（This example is taken from https://activecalculus.org/single/sec-2-3-prod-quot.html of https://activecalculus.org/single/.） Let $N(t)$ represent the number of shares owned on day $t$, where $t = 0$ represents the first day on which shares were purchased. Let $S(t)$ give the value of one share of the stock on day $t$; note that the units on $S(t)$ are dollars per share. How is the total value, $V(t)$, of the held stock on day $t$ determined?

Solution. Observe that the units on $N(t)$ are “shares” and the units on $S(t)$ are “dollars per share”. Thus when we compute the product

, it follows that the resulting units are “dollars”, which is the total value of held stock. Hence,

.

What is the meaning of the statement “$f(60) = 0.04$” in the context of the problem? That is, what does this say about the car's fuel economy? Write a complete sentence to explain.

Consider the function $g(v) = \frac{1}{f(v)}$. What is the value of $g(60)$? What are the units on $g$? What does $g$ measure?

Consider the function $h(v) = v \cdot f(v)$. What is the value of $h(60)$? What are the units on $h$? What does $h$ measure?

Do $f(60)$, $g(60)$, and $h(60)$ tell us fundamentally different information, or are they all essentially saying the same thing? Explain.

Suppose we also know that $f(70) = 0.045$. Find the average rate of change of $f$ on the interval $[60,70]$. What are the units on the average rate of change of $f$? What does this quantity measure? Write a complete sentence to explain.