# 3_7 - q • For each real zero r x = r is the vertical...

This preview shows pages 1–5. Sign up to view the full content.

Section 3.7: Rational Functions Instructor: Ms. Hoa Nguyen ([email protected]) 1 Rational Function A rational function f is a function of the form f ( x ) = p ( x ) q ( x ) where p ( x ) = a n x n + a n - 1 x n - 1 + ··· + a 1 x + a 0 ( a n 6 = 0) and q ( x ) = b m x m + b m - 1 x m - 1 + ··· + b 1 x + b 0 ( b m 6 = 0). The domain of a rational function f is the set of all x such that q ( x ) 6 = 0. Example 1 Example 2 1

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
2 Horizontal Asymptote The line of y = L is a horizontal asymptote of the graph of y = f ( x ) if f ( x ) L as x → ∞ or f ( x ) L as x → -∞ . The graph of f ( x ) = 1 ( x - 2) 2 . Finding horizontal asymptotes : Let f ( x ) = p ( x ) q ( x ) . The degree of p ( x ) is n and the degree of q ( x ) is m . If n < m (i.e., f is proper), y = 0 is the horizontal asymptote. If n = m , y = a n b m is the horizontal asymptote. Otherwise, NO horizontal asymptotes. Example 3 2
3 Vertical Asymptote The line of x = c is a vertical asymptote of the graph of y = f ( x ) if | f ( x ) | → ∞ as x c . The graph of f ( x ) = x +1 x - 1 . Note: If p ( x ) and q ( x ) of a rational function f ( x ) = p ( x ) q ( x ) have NO common factors, then f is said to be in the lowest term . Finding vertical asymptotes : Let f ( x ) = p ( x ) q ( x ) in the LOWEST term . Let q ( x ) = 0 to ﬁnd the real zeros of

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: q . • For each real zero r , x = r is the vertical symptote. Remark : If f is NOT in the LOWEST term, following the above steps may result in an incorrect listing of vertical asymptotes. Example 4 3 4 Oblique Asymptote The graph of f ( x ) = x 3 +2 x 2-1 x 2 + x +1 . Question : Based on the result of the long division, what line does f approach as | x | → ∞ ? Finding oblique asymptotes : ONLY when n = m + 1, there is an oblique asymptote which is the QUOTIENT computed in the long division. Otherwise, there is NO oblique asymptote. Example 5 5 Inverse Variation The number y varies inversely with the number x if y = k x for some k 6 = 0. In other words, y = f ( x ) where f ( x ) = k x for some k 6 = 0. 4 LEFT: f ( x ) = k x where k > 0. RIGHT: f ( x ) = k x where k < 0. 5...
View Full Document

## This note was uploaded on 05/23/2011 for the course MAC 1147 taught by Professor Nuegyen during the Spring '11 term at FSU.

### Page1 / 5

3_7 - q • For each real zero r x = r is the vertical...

This preview shows document pages 1 - 5. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online