2.3 polynomial and rational functions. polynomial and rational functions are often used to express...

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