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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 = x^{2} + 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.

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:

Because any vertical line would pass through only one point, y = x^{2} + 2 must be
assigning only one y value to each x value, and it therefore passes the vertical line
test. Thus, y = x^{2} + 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 = x^{2} + 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 x^{3}: