The vertical line test can be used to determine if an equation is a function. In order to be a function, there must only be one (or ) value for each value of . The vertical line test determines how many (or ) values are present for each value of . If a single vertical line passes through the graph of an equation more than once, it is not a function. If it passes through exactly once or not at all, then the equation is a function.

The horizontal line test can be used to determine if a function is one-to-one, that is, if only one value exists for each (or ) value. Calculating zeroes, domain, and range can be useful for graphing an equation, but they do not tell if it is a function.

Example of a function:

Example of an equation that is not a function:

A function is odd if and only if, for every in its domain, ; it is even if and only if, for every in its domain, . We can see that

It follows that is an odd function.

To identify a function as even, odd, or neither, determine by replacing with , then simplifying. If , the function is even; if is odd.

, so is an even function.

When determining how a the graph of a function will be translated, we know that anything that happens to x in the function will impact the graph horizontally, opposite of what is expressed in the function, whereas anything that is outside the function will impact the graph vertically the same as it is in the function notation.

For this graph:

The graph will move 3 spaces left, because that is the opposite sign of the what is connected to x directly.

Also, the graph will move down 2 spaces, because that is what is outside the function and the 2 is negative.

To identify a function as even, odd, or neither, determine by replacing with , then simplifying. If , the function is even; if is odd.

Since , is an odd function.

A function is odd if and only if, for every in its domain, ; it is even if and only if, for every in its domain, . We can see that

;

the function cannot be even. This does allow for the function to be odd. However, if is odd, then, by definition,

, or

and is equal to its own opposite - the only such number is 0, so

.

This is not the case - - so the function is not odd either.

As is clear from the graph, in the interval between ( included) to , the is constant at and then from ( not included) to ( not included), the is a decreasing function.

The relation graphed is a function, as it passes the vertical line test - no vertical line can pass through it more than once, as seen below:

Also, it is seen to be symmetrical about the -axis. This proves the function even.

A function is odd if and only if, for every in its domain, ; it is even if and only if, for every in its domain, . We can see that

Of course,

.

Therefore, is even by definition.

A function is odd if and only if, for every in its domain, ; it is even if and only if, for every in its domain, . We can see that

Of course,

.

Therefore, is even by definition.