Let us divide x 2 by (x + 1) by long division (or we can use synthetic division as well). Here the degree of numerator is 2 and that of denominator = 1. Its equation is y = quotient that is obtained by dividing the numerator by denominator using the long division.Д®xample: Find the slant asymptote of the function f(x) = x 2/(x+1). A rational function has a slant asymptote only when the degree of the numerator (N) is exactly one greater than the degree of the denominator (D). Slant (Oblique) Asymptotes of a Rational FunctionДЄ slant asymptote is also an imaginary oblique line to which a part of the graph appears to touch. Since N = D, the HA is y = (leading coefficient of numerator) / (leading coefficient of denominator) = 1/1 = 1. Here the degree of the numerator is, N = 2, and the degree of the denominator is, D = 2. If N = D, then the HA is y = ratio of the leading coefficients.Д®xample: Find the horizontal asymptote (if any) of the function f(x) = (x 2 + 5x + 6) / (x 2 + x - 2).Easy way to find the horizontal asymptote of a rational function is using the degrees of the numerator (N) and denominators (D). A rational function can have at most one horizontal asymptote. Here, 'some number' is closely connected to the excluded values from the range. Horizontal Asymptote of a Rational FunctionДЄ horizontal asymptote (HA) of a function is an imaginary horizontal line to which its graph appears to be very close but never touch. Thus, there is a VA of the given rational function is, x = 1. We have already seen that this function simplifies to f(x) = (x + 3) / (x - 1). Set the denominator = 0 and solve for (x) (or equivalently just get the excluded values from the domain by avoiding the holes).Д®xample: Find the vertical asymptotes of the function f(x) = (x 2 + 5x + 6) / (x 2 + x - 2).Simplify the function first to cancel all common factors (if any).

So to find the vertical asymptotes of a rational function: A rational function may have one or more vertical asymptotes. But note that there cannot be a vertical asymptote at x = some number if there is a hole at the same number. Here, 'some number' is closely connected to the excluded values from the domain.

Vertical Asymptote of a Rational FunctionДЄ vertical asymptote (VA) of a function is an imaginary vertical line to which its graph appears to be very close but never touch. Since (x + 2) was striked off, there is a hole at x = -2. Let us factorize the numerator and denominator and see whether there are any common factors.ДЇ(x) = / Holes exist only when numerator and denominator have linear common factors.Д®xample: Find the holes of the function f(x) = (x 2 + 5x + 6) / (x 2 + x - 2). Every rational function does NOT need to have holes. We can find the corresponding y-coordinates of the points by substituting the x-values in the simplified function. They can be obtained by setting the linear factors that are common factors of both numerator and denominator of the function equal to zero and solving for x. The holes of a rational function are points that seem that they are present on the graph of the rational function but they are actually not present. Apart from these, it can have holes as well. Now, we will solve this for x.ДЄ rational function can have three types of asymptotes: horizontal, vertical, and slant asymptotes.

Set of all real numbers other than the values of y mentioned in the last step is the range.Д®xample: Find the range of f(x) = (2x + 1) / (3x - 2).

Set the denominator of the resultant equation в‰ 0 and solve it for y.

If we have f(x) in the equation, replace it with y.

To find the range of a rational function y= f(x): The range of a rational function is the set of all outputs (y-values) that it produces. Thus, the domain = Range of Rational Function We set the denominator not equal to zero.

Set of all real numbers other than the values of x mentioned in the last step is the domain.Д®xample: Find the domain of f(x) = (2x + 1) / (3x - 2).

Set the denominator в‰ 0 and solve it for x.

To find the domain of a rational function y = f(x): The domain of a rational function is the set of all x-values that the function can take. This is the key point that is used in finding the domain and range of a rational function. Any fraction is not defined when its denominator is equal to 0.

Rational function graph