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.

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.

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.

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

Also, it can be seen to be symmetrical about the origin. Consequently, for each  in the domain,  - the function is odd.

The vertical asymptote(s) of the graph of a rational function such as  can be found by evaluating the zeroes of the denominator after the rational expression is reduced. The expression is in simplest form, so set the denominator equal to 0 and solve for :

The graph of  has the line of the equation  as its only vertical asymptote.

The vertical asymptote(s) of the graph of a rational function such as  can be found by evaluating the zeroes of the denominator after the rational expression is reduced. 

First, factor the numerator. It is a quadratic trinomial with lead term , so look to "reverse-FOIL" it as

by finding two integers with sum  and product 30. By trial and error, these integers can be found to be  and , so

Therefore,  can be rewritten as 

.

Cancelling , this can be seen to be essentially a polynomial function:

,

which does not have a vertical asymptote.

 is a rational function whose numerator and denominator have the same degree (1). As such, it has as a horizontal asymptote the line of the equation , where  is the quotient of the coefficients of the highest-degree terms of its numerator and denominator. Consequently, the horizontal asymptote of 

is 

,

or

.

 is a rational function whose numerator and denominator have the same degree (2). As such, it has as a horizontal asymptote the line of the equation , where  is the quotient of the coefficients of the highest-degree terms of its numerator and denominator. Consequently, the horizontal asymptote of 

is

or

.

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.

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

Also, it is seen to be symmetrical about the -axis. Consequently, for each  in the domain,  - the function is even.