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

It follows thatis an odd function.

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

;

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

, or

andis 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 functionis odd if and only if, for everyin its domain,; it is even if and only if, for everyin 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:

此外,它can be seen to be symmetrical about the origin. Consequently, for eachin the domain,- the function is odd.

The vertical asymptote(s) of the graph of a rational function such ascan 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 ofhas the line of the equationas its only vertical asymptote.

The vertical asymptote(s) of the graph of a rational function such ascan be found by evaluating the zeroes of the denominatorafter 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 sumand product 30. By trial and error, these integers can be found to beand, 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, whereis 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, whereis 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:

此外,它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:

此外,它is seen to be symmetrical about the-axis. Consequently, for eachin the domain,- the function is even.