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Properties of Rational Functions. Learning Objectives. 1. Find the domain of a rational function 2. Find the vertical asymptotes of a rational function 3. Find the horizontal or oblique asymptotes of a rational function. Rational Function. To Find the Domain. Domain
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Learning Objectives 1. Find the domain of a rational function 2. Find the vertical asymptotes of a rational function 3. Find the horizontal or oblique asymptotes of a rational function
To Find the Domain Domain The domain of a rational function is all real values except where the denominator, q(x) = 0
Examples Find holes and vertical asymptotes
Examples Find holes and vertical asymptotes
Example Find holes and vertical asymptotes
Holes and Vertical Asymptotes Holes and vertical asymptotes are discontinuities, but they are very different vertical asymptotes are non-removable discontinuities but holes are removable discontinuities, and by the addition of a point, we can create a function continuous at that point
Example Hole at x = 2 X-2 evenly divides both the numerator and the denominator Holes do not appear on the graph, but are clearly indicated on the table
Example Vertical Asymptote at x = 2 Holes do appear on the graph and are clearly indicated on the table
Example x 0- f(x) ∞ x 0+ f(x) ∞ x 0 f(x) ∞
End Behavior A function will not have both an oblique and a horizontal asymptote
Horizontal Asymptote A horizontal line is an asymptote only to the far left and the far right of the graph. "Far" left or "far" right is defined as anything past the vertical asymptotes or x-intercepts. Horizontal asymptotes are not asymptotic in the middle. It is okay to cross a horizontal asymptote in the middle.
Example Find equation for horizontal asymptote
Example Find equation for horizontal asymptote
Example Find x-value(s) where f(x) crosses horizontal asymptote
Example Find equation for horizontal asymptote
Example Find equation for horizontal asymptote
Example Find equation for horizontal asymptote
Example This means for very large values of R2 the total resistance approaches 10 ohms.
Oblique Asymptotes When the degree of the numerator is exactly one more than the degree of the denominator, the graph of the rational function will have an oblique asymptote. Another name for an oblique asymptote is a slant asymptote. To find the equation of the oblique asymptote, perform long division (synthetic if it will work) by dividing the denominator into the numerator and discarding the remainder
Example Oblique Asymptote y = x+2 X-2 divides the numerator with a remainder Y2 is the end behavior of y1
Example Using synthetic division
Example Our rational function Our rational function and OA
Example Using synthetic division
Example Our rational function Our rational function and OA