is a rational function in simplest form whose denominator is a polynomial with degree greater than that in its numerator (2 and 1, respectively). The graph of such a function has as its horizontal asymptote the line of the equation .

 is a rational function whose numerator and denominator are polynomials of 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 is a polynomial with degree greater than that of the polynomial in its denominator (2 and 1, respectively). The graph of such a function does not have a horizontal asymptote.

To find an oblique asymptote of a rational function whose numerator has higher degree than its denominator, as is the case here, divide the former by the latter, as follows:

Note that "missing" terms have been inserted in the dividend as terms with zero coefficients.

Divide the leading term of the dividend by that of the divisor:

Place this in the quotient, and multiply this by the divisor:

Subtract this from the dividend. The figure should look like this:

Repeat with the difference:

The figure now looks like this:

The difference has degree less than that of the divisor, so the division is finished. The equation of the oblique asymptote of the graph of  is taken from the quotient, and is the line of the equation .

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 and denominator.

The numerator is a perfect square trinomial and can be factored as such:

The denominator can be factored as the difference of squares:

Rewrite

as

The expression can be reduced by cancelling  in both halves:

Set the denominator equal to 0 and solve:

The only asymptote is therefore the line of the equation . 

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.

To find an oblique asymptote of a rational function whose numerator has higher degree than its denominator, as is the case here, divide the former by the latter, as follows:

Divide the leading term of the dividend by that of the divisor:

Place this in the quotient, and multiply this by the divisor:

Subtract this from the dividend. The figure should look like this:

Repeat with the difference:

The figure now looks like this:

The difference has degree less than that of the divisor, so the division is finished. The oblique asymptote is the quotient, 

The domain includes the values that go into a function (the x-values) and the range are the values that come out (the  or y-values). A sine curve represent a wave the repeats at a regular frequency. Based upon this graph, the maximum  is equal to 1, while the minimum is equal to –1. The x-values span all real numbers, as there is no limit to the input fo a sine function. The domain of the function is all real numbers and the range is .

A function has to pass the vertical line test, which means that a vertical line can only cross the function one time.  To put it another way, for any given value of , there can only be one value of .  For the function , there is one value for two possible  values.  For instance, if , then .  But if , as well.  This function fails the vertical line test.  The other functions listed are a line,, the top half of a right facing parabola, , a cubic equation, , and a semicircle, . These will all pass the vertical line test.