Pre Calculus Rational Functions & Asymptotes

Rational Function Rational functions are quotients of polynomial functions. This means rational ( ) functions can be expressed as ( ) where p and q are polynomial ( ) functions and q(x) is not zero. The domain of a rational function is the set of all real numbers except the x-values that make the denominator zero.

Example: x  5 (a) Find the domain of fx   . x2  25

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x2  25 (b) Find the domain of fx   . x 5

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x (c) Find the domain of gx   . x2  25

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over Pre Calculus (2.6) Rational Functions & Asymptotes

Notes on Notation Symbol Meaning Symbol Meaning xa  x approaches a from the right side xa  x approaches a from the left side

x  x approaches infinity x   x approaches negative infinity

Definition of a Vertical Asymptote The line xa is a vertical asymptote of the graph of a function f if fx  increases or decreases without bound as x approaches a. Symbolically this means that, fx()  as xa  or as xa  .

Locating Vertical Asymptotes px  If fx   is a rational function in which px  and qx  have no common qx  factors and a is a zero of qx  , the denominator, then xa is a vertical asymptote of the graph of f.

CHECK THE RATIONAL FUNCTION FOR HOLES (when a term divides out from both the numerator and denominator). Examples: x (a) Find the vertical asymptote for gx   . x2  25

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2xx2  2 4 (b) Find the vertical asymptote for fx   . x2 9

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3x2  12 (c) Find the vertical asymptote for hx   . xx2 23

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Pg 2 over Pre Calculus (2.6) Rational Functions & Asymptotes

x  2 (d) Find the vertical asymptote for fx   . x2  4

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There is a hole at (2, ¼) and a vertical asymptote at x = – 2

sin x (e) Find the vertical asymptote for hx   . x3

x (f) Find the vertical asymptote for hx   . xx2 

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There is a hole at (0, – 1) and a vertical asymptote at x = 1

Pg 3 over Pre Calculus (2.6) Rational Functions & Asymptotes

Definition of a Horizontal Asymptote nn1 Nx  ann x a11 xa x  0 a  Let f be the rational function: fx  mm1 D x bmm x b11 xb x 0 b

The line yb is a horizontal asymptote of the graph of a function f if fx  approaches b as x increases or decreases without bound. Symbolically this means

that, f() x b1 as x  or f() x b2 as x  .

Strategies for finding Horizontal Asymptotes The degree of the numerator is n. The degree of the denominator is m. 1. If nm , the x-axis, or y  0, is the horizontal asymptote of the graph of f.

2. If nm , the line yab nm is the horizontal asymptote of the graph of f. 3. If nm , the graph of f has no horizontal asymptote.

Examples: 3 (a) Find any horizontal asymptotes for cx   . x 1

3x (b) Find any horizontal asymptotes for fx   . x2 1

3x2 (c) Find any horizontal asymptotes for gx   . x2 1

3x3 (d) Find any horizontal asymptotes for hx   . x2 1

Pg 4 over Pre Calculus (2.6) Rational Functions & Asymptotes

x 10 (e) Find any horizontal asymptotes for tx   . x  2

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a b c Table of values graph expanded graph c

sin x (f) Find any horizontal asymptotes for tx   . x3

Pg 5 Pre Calculus (2.6) Rational Functions & Asymptotes

Applications Example: (a) A drug is administered to a patient and the concentration of the drug in the bloodstream is monitored. At time t  0 (in hours since giving the drug), the 5t concentration (in mg/L) is given by ct   . Graph the function ct  t 2 1 and include any asymptote(s). (1) What is the highest concentration of drug that is reached in the patient’s bloodstream? (2) What happens to the drug concentration after a long period of time? (3) How long does it take for the concentration to drop below 0.3 mg/L?

(1) 2.5 mg/L (2) n=1 and m = 2, n

(3)

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16.6 hours

Pg 6