Precalculus: An Active Reading Approach: Elementary Functions

Section 8.1 Combinations of Functions

There are many ways and reasons to combine two or more functions together. In the first exercise, you will combine two functions in the context of performing a familiar activity: paying bills

If you need to add two functions together, such as \(f(x)+g(x)\text{,}\) this represents a combination of functions.

This combination is actually a new function of \(x\text{,}\) whose output is just the sum of the outputs of the functions \(f\) and \(g\text{.}\)

Of course, there are other ways to combine the outputs of two or more functions. In the next exercise, you will practice evaluating other function combinations.

Again, combining functions just means “doing arithmetic” on the outputs of functions. In context, we should be able to describe the meaning of different function combinations.

Now we look at the graphs of function combinations. In the next exercise, you will see the effect of multiplying the outputs of two functions to create the combination:

By now, you know that all combinations of functions are really just doing arithmetic on the the outputs of the functions. From a graphical standpoint, this just involves combining the \(y\) values of points on the graphs of \(f\) and \(g\text{.}\)

The next problem will have you explore the graph of a particular function combination:

Remember that this is all about combining the outputs of the \(f\) and \(g\) functions.