Section 7.1 Combining Functions

Supplemental Videos.

The main topics of this section are also presented in the following videos:

Subsection 7.1.1

In previous chapters, we have discussed composition of functions. Composing functions is one example of a way to combine two functions to create a new function. Some common ways to combine two functions $f(x)$ and $g(x)$ would be:

\begin{equation*} f(g(x))\ ,\ \ f(x)\pm g(x)\ ,\ \ \frac{f(x)}{g(x)}\ ,\ \text{and}\ f(x)\cdot g(x). \end{equation*}

However, it is very important to think about compatibility of two functions before we start combining them in various ways. For example, to combine the functions $f(x)$ and $g(x) $ as

\begin{equation*} \frac{f(x)}{g(x)} \end{equation*}

we have to know that $g(x)\neq 0 $ as we cannot divide by 0.

Furthermore, a compatability issue that we need to think about with all combinations of functions is with their domain and range. For, in each combination other than composition, we are assuming that the two functions have the same domain. So, if we are plugging in an amount in dollars to $f(x)$ and plugging in a type of vegetable to $g(x)$ it would not make sense to combine these two functions using addition, subtraction, or multiplication. Similarly, if the output of $f$ is a type of fruit and the output of $g$ is a year, then we cannot combine the functions as adding, dividing, or multiplying a fruit and a year doesn’t make sense.

This section will be all about taking a real world example and creating a function out of the information provided to you. The difficult part of the problems will be making sure that the combinations made make sense and give you the desired information when combined in the way you chose.

Previously, we gave some tips for solving word problems. Let’s take another look at this list (with the parts about exponential functions removed):

Tips for Starting Word Problems.

Identify the units of the input and output of the function given, or the function you are instructed to make.
If you are told to make a function, write down the general form of the function you are asked to make.
Plug in all given information, and think about what exactly you are being instructed to do.
Once you have the final answer, go back to the problem and make sure your answer makes sense in the context of the problem.

Example 7.2.

An apartment complex needs to relocate its residents due to water damage after a large storm. All the residents belongings are being packed into moving boxes of the same size. A moving company has been hired to relocate the individuals in the apartment building to their new location. The company has found that, on average, if they are relocating $x $ people then they will need $B(x) $ boxes. The company can only fit $150 $ boxes into one of their trucks. Write a function which tells the company how many trucks they will need to move $x $ people.

Solution.

As suggested above, let’s begin by listing the units of the input and output of the functions given and the function we are to create. The function we are given, $B(x)\text{,}$ has as its input a number of people and outputs the number of boxes needed.

The function we are to create needs an input of people and output of trucks needed, let’s call this function $T(x)\text{.}$ Since $B(x) $ is the number of boxes, and each truck can hold 150 boxes, the number of trucks needed should be

\begin{equation*} T(x)=\frac{B(x)}{150}. \end{equation*}

Note that this function makes sense as the input of $B(x)$ is number of people which is also what we wanted our function $T(x)$ to have as an input.

Exercises 7.1.2 Exercises

1. Combining Formulas.

Let $f(x)=x^3-3$ and $g(x)=x^4+5\text{.}$ Find the following functions:

a) $f(x)+g(x)=$

b) $f(x)g(x)=$

c) $f(x)-g(x)=$

d) $\dfrac{f(x)}{g(x)}=$

Hint.

For each part, replace "$f(x)$" and "$g(x)$" with the given formulas everywhere they appear.

2. Profit, Revenue, and Cost.

It costs a golf course $C(h) = 200 h$ dollars to operate for $h$ hours. Each hour they are open, they bring in $ 260.

a) Write a function for the revenue, $R(h)\text{,}$ as a function of the number of hours they are open.

$R(h)=$

b) Find a function $P(h)$ for the profit of the golf course after being open for $h$ hours.

$P(h)=$

Hint.

a) Revenue is the total amount of money earned. The golf course brings in 260 dollars every hour. Use this to write a formula for the revenue (it should be a linear function).

b) Recall that to determine profits, you need to take the total amount of money earned and subtract the costs. In other words, profit$=$ revenue-cost.

3. Writing and Interpretting Combined Functions.

A restaurant chain is considering raising the prices on their burgers. They have found that for every $ 1 increase, their customers will buy 50 fewer burgers each day. Say that the price of a burger is currently $ 6, and their customers buy 650 burgers per day.

a) Write a formula $p(x)\text{,}$ which gives the price per burger (in dollars) after an increase of $x$ dollars per burger.

$p(x)=$

b) Write a formula $k(x)\text{,}$ which represents the number of burgers the chain sells each day after an increase of $x$ dollars per burger.

$k(x)=$

c) Which of the following gives an interpretation of $p(x) \cdot k(x)$ in real-world terms.

$p(x)\cdot k(x)$ gives the daily revenue of the restaurant after an increase of $x$ dollars per burger.
$p(x)\cdot k(x)$ gives the increase in profit after an increase of $x$ dollars per burger.
$p(x)\cdot k(x)$ gives the daily profit of the restaurant after an increase of $x$ dollars per burger.
$p(x)\cdot k(x)$ gives the total price of a burger after an increase of $x$ dollars per burger.

d) Find the approximate value of $x$ which maximizes $p(x) \cdot k(x)\text{.}$

$x=$ dollars

Hint.

a) Since $p(x)$ gives the price per burger after increasing the price by $x$ per burger, we need to take the initial price per burger and add $x$ to get the new price.

b) Since the restaurant chain will sell 50 fewer burgers for every dollar increase in price per burger ($x$), we need to take the current number of burgers the chain sells and subtract 50 for every increase of $x\text{.}$ Think about how to write this as a linear function.

c) $p(x)$ gives the price per burger after an increase of $x$ dollars per burger and $k(x)$ gives the total number of burgers the chain sells after an increase of $x$ dollars per burger. Think about what multiplying these two values together will give you.

d) $p(x)\cdot k(x)$ is a quadratic function. Recall that the maximum value of a quadratic function occurs at it’s vertex. There are several ways to find the vertex of a quadratic function, but we can use information about the horizontal intercepts to find our answer. Note that the vertex occurs half-way between the horizontal intercepts.

4. Writing Formulas.

Mary Beth has a farm, and she keeps only sheep and horses. Let $S(t)$ and $H(t)$ stand for the number of sheep and horses, respectively, on Mary Beth’s farm in year $t\text{.}$

a) Write functions $P_S(t)$ and $P_H(t)\text{,}$ which give the percentage of sheep and horses, respectively, out of the total number of animals on the farm.

$P_S(t)=$

$P_H(t)=$

b) Evaluate $P_S(t) + P_H(t)\text{.}$

$P_S(t) + P_H(t) =$

Hint.

a) In order to find the percentage of sheep and horses out of the total number of animals on the farm we need to do the following.

To find the percentage of sheep, we will take $\dfrac{\text{total number of sheep}}{\text{total number of animals}}.$ Similarly, to find the percentage of horses, we will take $\dfrac{\text{total number of horses}}{\text{total number of animals}}.$

The number of sheep is given by $S(t)\text{,}$ the number of horses is given by $H(t)\text{.}$ Since Mary Beth only has sheep and horses on her farm, the total number of animals is given by the sum of the number of sheep and the number of horses: $S(t)+H(t)\text{.}$

b) To evaluate $P_S(t)+P_H(t)\text{,}$ we want to add the formulas we found above.

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