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SectionCombining Functions

In sections 6.2 and 9.1 we 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 compatability 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 combatability issue that we need to think about with all combinations of functions is with their domain and range. For, in each combination, 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. 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.

Back in section 8.4 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
  1. Identify the units of the input and output of the function given, or the function you are instructed to make.

  2. If you are told to make a function, write down the general form of the function you are asked to make.

  3. Plug in all given information, and think about what exactly you are being instructed to do.

  4. Once you have the final answer, go back to the problem and make sure your answer makes sense in the context of the problem.

Example297

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.