Rational Functions

Functions of the form:

Note:

· Most of our examples assume f(x) and g(x) do not contain any common factors (other than constants). If f(x) and g(x) do contain common factors find the domain first, reduce second!

Consider

1. Domain: All real numbers for which

2. x-intercept(s) (Let y = 0 and solve for x. The x values will be values of x that make the numerator = 0 (if there are any).)

3. y-intercept (Let x = 0 and solve for y.)

4. Asymptotes: Lines that break up the graph.

· Vertical: The line x = c, where the value c makes the function undefined, i.e. g(c)=0 in the above definition. (after it has been reduced to lowest terms, if necessary). In this case as x gets closer and closer to the value c, the y value gets closer and closer to either .

· Horizontal: The line y = k, such that as x gets closer and closer to the y value gets closer and closer to k.

The horizontal asymptote (if there is one) is found via

a. Look for the highest degree term in the denominator.

b. Divide every term by the term you chose in part a.

c. Take the limit as x.

d. If the expression approaches a constant value, call it k, then the horizontal asymptote is the line y = k, if not then there is no horizontal asymptote.

· Slant or Oblique Asymptote: The line y = quotient (found via long division if degree of the numerator > degree of the denominator). See exercise 41.

5. Sign Chart to determine behavior – often useful if we do not use translation.

· Plot all x-intercepts and vertical asymptote values on the sign chart

· Use this to determine behavior of the function.

6. Keep in mind special circumstances we may eventually encounter such as:

· Some graphs actually cross a horizontal asymptote

· Some graphs have holes. Functions, which can be reduced to a simpler polynomial often, have holes.