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 may only have one y-value for each x-value.

The vertical line test can be used to identify the function. If at any point on the graph, a straight vertical line intersects the curve at more than one point, the curve is not a function.

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.

Perpendicular lines have slopes that are negative reciprocals.  This is the case with these two lines.  Although these lines interesect, this is not the most appropriate answer since it does not account for the fact that they are perpendicular.

To find the slope of an equation first get the equation in slope intercept form.

where,

 represents the slope.

Thus

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.

,

so

By the Power of a Product Property,

,

so  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

By the Power of a Product Property,

, so  is not an even function.

,

, so  is not 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.

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.