CK-12-Pre-Calculus Concepts

http://www.ck12.org Chapter 2. Polynomials and Rational Functions

Guided Practice

1. Identify the horizontal asymptotes of the following function.

f(x) =|((xx−− 53 ))(|·x(x+−^21 ))

2. Identify the vertical and horizontal asymptotes and the holes of the following function.

f(x) =(x

(^4) − 9 )(x− 1 )
(x^2 + 3 )(x− 3 )

3. Identify the horizontal asymptotes if they exist for the following 3 functions.

a. f(x) =^3 x

(^6) − 72 x
x^6 + 999
b.h(x) =ax
(^4) +bx (^3) +cx (^2) +dx+e
f x^4 +gx^3 +hx^2
c.g(x) =hf((xx))
Answers:

1. First notice the absolute value surrounding one of the terms in the denominator. The degrees of both the numerator
   and the denominator will be 2 which means that the horizontal asymptote will occur at a number. Asxgets infinitely
   large, the function is approximately:

f(x) =x

2

x^2

So the horizontal asymptote isy=−1 asxgets infinitely large.
On the other hand, asxgets infinitely small the function is approximately:

f(x) = x

2

−x^2

So the horizontal asymptote isy=−1 asxgets infinitely small.
In this case, you cannot blindly use the leading coefficient rule because the absolute value changes the sign.

2. The numerator of the function factors to be:

f(x) =(x

(^2) + 3 )(x (^2) − 3 )(x− 1 )
(x^2 + 3 )(x− 3 ) =
(x^2 − 3 )(x− 1 )
x− 3
Note that a factor does cancel and also notice that this factor is never equal to zero. Not all factors that cancel
indicate a hole. A horizontal asymptote does not exist because the degree of the numerator is greater than the degree
of the denominator. The vertical asymptote is atx=3.
3.