2.3 Polynomial and 2.3 Polynomial and Rational FunctionsRational Functions
Polynomial and rational functions are often used to express relationships in application problems.
DEFINITION:
The line x = a is a vertical asymptote if any of the following limit statements are true:
limx a
f x limx a
f x
limx a
f x .limx a
f x
•If a makes the denominator zero, but doesn’t make the numerator zero, then x = a is a vertical asymptote.
•If a makes both the denominator and the numerator zero, then there is a hole at x=a
Example 2: Determine the vertical asymptotes of the function given by
f (x) x(x 2)
x(x 1)(x 1)
f (x) (x 2)
(x 1)(x 1)
• Since x = 1 and x = –1 make the denominator 0, but don’t make the numerator 0, x = 1 and x = –1 are vertical asymptotes.
• x=0 is not a vertical asymptote since it makes both the numerator and denominator 0.
The line y = b is a horizontal asymptote if either or both of the following limit statements are true:
orlimx
f x b limx
f x b.
The graph of a rational function may or may not cross a horizontal asymptote. Horizontal asymptotes occur when the degree of the numerator is less than or equal to the degree of the denominator.
Same: y = leading coefficient/leading coefficientBOB: y = 0TUB: undefined (no H.A.)
f (x) 3x2 2x 4
2x2 x 1.
Determine the horizontal asymptote of the function given by
Example of holeExample of hole
Figure 45Figure 45
Intercepts. The x-intercepts occur at values for which y = 0. For a fraction to = 0, the numerator must equal 0. Since 8 ≠ 0, there are no x-intercepts. To find the y-intercept, let x = 0.
y-intercept (0, 8/5)
3x-5
8y of intercepts theFind
5
8y
Suppose the average cost per unit in dollars, to produce x units of a product is given by
30
500
x
xC
)10(C )50(C )100(C(a) find
(b) Graph the function and identifyany intercepts and asymptotes
C