Function Notation

The first thing to examine in this unit is new notation. You have already explored what makes a function a function in the Relations and Functions Unit. Now you will examine some new notation that goes hand in hand with functions: Function Notation.

Think of a function as a machine. You plug in a value for x, and out pops one (and only one) value for y. For example, look at the following function:

y = x² + 2x + 1

If we plug in x = 2, we get:

y = (2)² + 2(2) +1
y = 4 + 4 + 1
y = 9

Every time we plug in x = 2, we get 9 NO MATTER WHAT. This is the beauty of functions. Now, instead of relying on y (which really could be any variable) we could say:

The value of the function x² + 2x + 1 evaluated at x = 2 is 9.

Now mathematicians are fairly lazy people! We don't want to say all of those words everytime we talk about evaluating a function at a given value of x. This is where function notation comes in. Function notation looks like this:

f(x)

The above statement, read in english, says "The value of function f, evaluated at x" This notation replaces the y, and is commonly referred to as F of X. Let's look at how this works with our example. Re-writing our function using function notation we get:

f(x) = x² + 2x + 1

To evaluate the function at x = 2, you write f(2). (The value of the function f evaluated at x = 2). The algebra is as follows:

f(2) = (2)² + 2(2) +1
f(2) = 4 + 4 + 1
f(2) = 9

ONE of the many reasons this notation is handy is that in the statement f(2) = 9 you get to see what the value of x was (2) as well as the result of the function (9) - as opposed to ONLY seeing the result of the function when you are looking at y = 9.

A quick note: a function does not need to be named f. You can use ANY letter to name a function, but usually you will see letters starting at f and working upwards: For example you might see f(x), g(x), h(x) and so on.

Now that you have seen an introduction to function notation, let's play a bit:

Example 1

Given the function f(x) = x³, evaluate f(3).

Solution

What this question is asking you to do is to find the value of function f (which is x³) at an x-value of 3. You should get the following:

f(3) = (3)³
f(3) = 27

Example 2

Given the function g(x) = x² + x - 7, evaluate g(z + 1).

Solution

You will not always be asked to evaluate a function at a number. In this example, just substitute (z + 1) in the function whereever you see an x:

g(z + 1) = (z + 1)² + (z + 1) - 7
g(z + 1) = z² + 2z + 1 + z + 1 - 7
g(z + 1) = z² + 3z - 5

Example 3

Given f(x) = x + 1, and g(x) = x² - 2, find f(g(3))

Solution

In this example, you have nested functions. This is called function composition. The solution is similar to what you have already seen: Just work from the 'inside' out. (This means evaluate g(3) first, then work on function f).

g(3) = (3)² - 2
g(3) = 9 - 2
g(3) = 7

Now since g(3) is equal to 7, evaluate f(g(3)) which is f(7).

f(7) = (7) + 1
f(7) = 8

Therefore your final solution is:

f(g(3)) = 8

After examining the work here, you can work on some sample questions, or head back to the polynomial and rational functions main page.