If you're reading this guide now, you've probably dealt with functions in great detail already, so I'll just include some brief highlights you'll need to get started with calculus. Much of this should be review, so feel free to skip sections you feel comfortable with.

Definition of a Function

A function is a rule that assigns to each element x from a set known as the "domain" a single element y from a set known as the "range". For example, the function y = x2 + 2 assigns the value y = 3 to x = 1, y = 6 to x = 2, and y = 11 to x = 3. Using this function, we can generate a set of ordered pairs of (x, y) including (1, 3),(2, 6), and (3, 11). We can also represent this function graphically, as shown below.

Figure %:Graph of the function y = x2 + 2

The Vertical Line Test

Note that in the graph above, each element x is assigned a single value y. If a rule assigned more than one value y to a single element x, that rule could not be considered a function. As you may recall from precalc, we can test for this property using the vertical line test, where we see whether we can draw a vertical line that passes through more than one point on the graph:

Figure %:Vertical line test on the function y = x2 + 2

Because any vertical line would pass through only one point, y = x2 + 2 must be assigning only one y value to each x value, and it therefore passes the vertical line test. Thus, y = x2 + 2 can rightfully be considered a function.

The Horizontal Line Test

Although a function can only assign one y value to each element x, it is allowed to assign more than one x value to each y. This is the case with our function y = x2 + 2. The value x = 4 is mapped to the single value y = 18, but the value y = 18 is mapped to both x = 4 and x = - 4.

A one-to-one function is a special type of function that maps a unique x value to each element y. So, each element x maps to one and only one element y, and each element y maps to one and only one element x. An example of this is the function x3: