# Evaluation and Composition of Functions

We have been using the function notation*f*(

*x*) intuitively; we also need to study what it actually means. You can think of the letter

*f*in the function notation

*f*(

*x*) as the name of the function. Instead of using the equation

*y*=

*x*

^{3}– 1 to describe the function, we can write

*f*(

*x*) =

*x*

^{3}– 1. Here,

*f*is the name of the function and

*f*(

*x*) is the value of the function at

*x*. So

*f*(2) = 2

^{3}– 1 = 8 – 1 = 7 is the value of the function at 2. As you can see, this notation affords a convenient way of prompting the evaluation of a function for a particular value of

*x*.

Any letter can be used as the independent variable in a function. So the above function could be written

*f*(*p*) =*p*^{3}– 1. This indicates that the independent variable in a function is just a “placeholder.” The function could be written without a variable as follows:

In this form, the function can be viewed as an input/output operation. If 2 is put into the function

*f*(2), then 2^{3}– 1 is returned.In addition to plugging numbers into functions, we can plug expressions into functions. Plugging

*y*+ 1 into the function yields

You can also plug other expressions in terms of

*x*into a function. Plugging 2*x*into the function yields

This evaluation can be troubling to students because the variable

*x*in the function is being replaced by the same variable. But the*x*in function is just a placeholder. If the placeholder were removed from the function, the substitution would appear more natural. In , we plug 2*x*into the left side*f*(2*x*) and it returns the right side .

# Composition

We have plugged numbers into functions and expressions into functions; now let’s plug in other functions. Since a function is identified with its expression, we have actually already done this.In the example above with and 2

*x*, let’s call 2

*x*by the name

*g*(

*x*). In other words,

*g*(

*x*) = 2

*x*. Then the composition of

*f*with

*g*(that is plugging

*g*into

*f*) is

You probably won’t see the notation

For another example, let and let . Then and .

*f*(*g*(x)) on the test. But you probably will see one or more problems that ask you perform the substitution.For another example, let and let . Then and .

Once you see that the composition of functions merely substitutes one function into another, these problems can become routine.

Notice that the composition operation

A function can also be composed with itself. That is, substituted into itself. Let . Then .

Notice that the composition operation

*f*(*g*(*x*)) is performed from the inner parentheses out, not from left to right. In the operation*f*(*g*(2)), the number 2 is first plugged into the function*g*and then that result is plugged in the function*f*.A function can also be composed with itself. That is, substituted into itself. Let . Then .

Example

The graph of *y* = *f*(*x*) is shown to the right. If *f*(–1) = *v*, then which one of the following could be the value of *f*(*v*) ?

- 0
- 1
- 2
- 2.5
- 3

Since we are being asked to evaluate *f*(*v*) and we are told that *v* = *f*(–1), we are just being asked to compose f(*x*) with itself.

That is, we need to calculate *f*(*f*(–1)).

From the graph, *f*(–1) = 3. So *f*(*f*(–1)) = *f*(3).

Again, from the graph, *f*(3) = 1. So *f*(*f*(–1)) = *f*(3) = 1.

The answer is (B).