5. Rational Functions

Introduction to Rational Functions

Multiple Choice
Find the domain of the rational function. Then, write it in lowest terms.
$f\left(x\right)=\frac{x^2+9}{x-3}$
Multiple Choice
Find the domain of the rational function. Then, write it in lowest terms.
$f\left(x\right)=\frac{6x^5}{2x^2-8}$
Open Question
Provide a short answer to each question. What is the domain of the function ƒ(x)=1/x? What is its range?
Open Question
Provide a short answer to each question. Is ƒ(x)=1/x^2 an even or an odd function? What symmetry does its graph exhibit?
Open Question
In Exercises 1–8, find the domain of each rational function. f(x)=(x+7)/(x^2+49)
Open Question
Use the graph of the rational function in the figure shown to complete each statement in Exercises 15–20.
As x -> -2^+, f(x) -> __
Open Question
In Exercises 21–36, find the vertical asymptotes, if any, and the values of x corresponding to holes, if any, of the graph of each rational function. f(x)=x/(x+4)
Open Question
Match the rational function in Column I with the appropriate descrip-tion in Column II. Choices in Column II can be used only once. ƒ(x)=(x^2-16)/(x+4)
Open Question
In Exercises 21–36, find the vertical asymptotes, if any, and the values of x corresponding to holes, if any, of the graph of each rational function. r(x)=(x^2+4x−21)/(x+7)
Open Question
In Exercises 37–44, find the horizontal asymptote, if there is one, of the graph of each rational function. f(x)=12x/(3x^2+1)
Open Question
Give the equations of any vertical, horizontal, or oblique asymptotes for the graph of each rational function. See Example 4. ƒ(x)=(x^2+1)/(x^2+9)
Open Question
In Exercises 45–56, use transformations of f(x)=1/x or f(x)=1/x^2 to graph each rational function. h(x)=1/x2 − 4
Open Question
In Exercises 55–56, use transformations of f(x) = (1/x) or f(x) = (1/x^2) to graph each rational function. g(x) = 1/(x + 2)^2 - 1
Open Question
Identify any vertical, horizontal, or oblique asymptotes in the graph of y=ƒ(x). State the domain of ƒ.
Open Question
In Exercises 57–80, follow the seven steps to graph each rational function. f(x)=(3x^2+x−4)/(2x^2−5x)