Properties of Rational Functions 1

Learning Objectives 2 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 The domain of a rational function is all real values except where the denominator, q(x) = 0

Example

Points Not in The Domain

Holes and Vertical Asymptotes

Examples Find holes and vertical asymptotes

Examples Find holes and vertical asymptotes

Example Find holes and vertical asymptotes

Example

More on Holes 14

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 15

Example Hole at x = 2 Holes do not appear on the graph, but are clearly indicated on the table X-2 evenly divides both the numerator and the denominator 16

Example Vertical Asymptote at x = 2 Holes do appear on the graph and are clearly indicated on the table 17

Example x  0 -  f(x)  ∞ x  0 +  f(x)  ∞ x  0  f(x)  ∞

Horizontal and Oblique Asymptotes

End Behavior A function will not have both an oblique and a 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. Horizontal Asymptote

Example Find equation for horizontal asymptote 22

Example Find equation for horizontal asymptote 23

Example Find x-value(s) where f(x) crosses horizontal asymptote 24

Example Find equation for horizontal asymptote 25

Example Find equation for horizontal asymptote 26

Find equation for horizontal asymptote Example 27

Example 28

29 Example This means for very large values of R 2 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 30

Example Oblique Asymptote y = x+2 Y2 is the end behavior of y1 X-2 divides the numerator with a remainder 31

Finding Oblique Asymptotes 32

Example 33

Example 34

Example 35

Example 36

Example Using synthetic division 37

Our rational function Our rational function and OA Example 38

Example 39

Example Using synthetic division 40

Our rational function Our rational function and OA Example 41