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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  = x3 – 1 to describe the function, we can write f(x) = x3 – 1. Here, f is the name of the function and f(x) is the value of the function at x. So f(2) = 23 – 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) = p3 – 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 23 – 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 2x 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 2x into the left side f(2x) and it returns the right side .


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 2x, let’s call 2x by the name g(x). In other words, g(x) = 2x. Then the composition of f with g (that is plugging g into f) is


You probably won’t see the notation 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 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 .

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) ?

  1. 0
  2. 1
  3. 2
  4. 2.5
  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).

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