Rational Functions and Their Graphs

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Questions and Answers

Which of the following represents a rational function?

$k(x) = 3x^3 - 5x$

$h(x) = x^2$

$g(x) = \frac{1}{x + 1}$ (correct)

$f(x) = 2x + 1$

The function $f(x) = \frac{1}{x}$ has a domain that includes 0.

False

What is the domain of the function $g(x) = \frac{1}{x + 1}$?

x = {x | x ≠ -1}

The range of the function $f(x) = \frac{x}{x - 1}$ is __________.

y = {y | y ≠ 1} Signup and view all the answers

Match the following functions with their corresponding domain:

$f(x) = \frac{1}{x}$ = $x ≠ 0 $g(x) = \frac{1}{x + 1}$ = $x ≠ -1 $F(x) = \frac{x}{x - 1}$ = $x ≠ 1 $G(x) = \frac{2x}{x + 3}$ = $x ≠ -3 Signup and view all the answers

What is the common feature of the graphs of the functions $f(x)$ and $g(x)$?

They do not touch the x-axis. Signup and view all the answers

Identify the range of the function $f(x) = \frac{1}{x}$.

y = {y | y ≠ 0} Signup and view all the answers

For the function $G(x) = \frac{2x}{x + 3}$, the value that x cannot take is __________.

-3 Signup and view all the answers

What is the vertical asymptote of the function $g(x) = \frac{2x - 1}{x + 3}$?

x = -3 Signup and view all the answers

The horizontal asymptote of the function $g(x) = \frac{2x - 1}{x + 3}$ is y = 2.

True Signup and view all the answers

Identify one point on the graph of the function $g(x) = \frac{2x - 1}{x + 3}$ when x = -5.

11/2 Signup and view all the answers

The equation of the horizontal asymptote is __________.

y = 2 Signup and view all the answers

Match the following functions with their descriptions:

f(x) = \frac{x - 2}{x + 5} = Rational function with a linear numerator and denominator F(x) = \frac{2 - 3x}{x - 3} = Rational function decreasing as x increases g(x) = \frac{1 - x}{x + 4} = Rational function with a negative slope G(x) = \frac{3x}{x - 3} = Rational function with a vertical asymptote at x = 3 Signup and view all the answers

What is the domain of the function defined by $f(x) = \frac{x - 2}{x - 3}$?

x = {x | x ≠ 3} Signup and view all the answers

The range of the function $g(x) = \frac{2x - 1}{x + 3}$ includes the value 2.

False Signup and view all the answers

Identify the vertical asymptote for the function $g(x) = \frac{2x - 1}{x + 3}$.

x = -3 Signup and view all the answers

The vertical asymptote of the function $f(x) = \frac{x - 2}{x - 3}$ is __________.

x = 3 Signup and view all the answers

Match the following asymptotes with their corresponding functions:

$f(x) = \frac{x - 2}{x - 3}$ = Vertical Asymptote: x = 3 $g(x) = \frac{2x - 1}{x + 3}$ = Vertical Asymptote: x = -3 $h(x) = \frac{x + 1}{x - 4}$ = Vertical Asymptote: x = 4 $k(x) = \frac{x^2 - 1}{x + 2}$ = Vertical Asymptote: x = -2 Signup and view all the answers

What does the range of the function $f(x) = \frac{x - 2}{x - 3}$ exclude?

2 Signup and view all the answers

The graph of $g(x) = \frac{2x - 1}{x + 3}$ will intersect the line y = 2.

False Signup and view all the answers

For the function $g(x) = \frac{2x - 1}{x + 3}$, what is the horizontal asymptote?

y = 2 Signup and view all the answers

Study Notes

Rational Functions Overview

A rational function is expressed as ( r(x) = \frac{p(x)}{q(x)} ), where ( p(x) ) and ( q(x) ) are polynomial functions and ( q(x) \neq 0 ).

Domain and Range

The domain of a rational function excludes values that make the denominator zero.
The range typically consists of all real numbers except specific values determined by vertical asymptotes.

Example Functions

For ( f(x) = \frac{1}{x} ):
- Domain: ( x \in {x | x \neq 0} )
- Range: ( y \in {y | y \neq 0} )
For ( g(x) = \frac{1}{x + 1} ):
- Domain: ( x \in {x | x \neq -1} )
- Range: ( y \in {y | y \neq 0} )
For ( F(x) = \frac{x}{x - 1} ):
- Domain: ( x \in {x | x \neq 1} )
- Range: ( y \in {y | y \neq 1} )
For ( G(x) = \frac{2x}{x + 3} ):
- Domain: ( x \in {x | x \neq -3} )
- Range: ( y \in {y | y \neq 2} )

Asymptotes

Vertical asymptotes occur at values excluded from the domain.
Horizontal asymptotes relate to the range where function output approaches certain values but never meets them.

Graphing Guidelines

Identify and plot key asymptotes.
Select values to the left and right of vertical asymptotes for more accurate graphing.
Extend curves approaching but not crossing asymptotes, reflecting symmetry or mirroring in behavior.

Specific Examples of Asymptotes

For ( f(x) = \frac{x - 2}{x - 3} ):
- Vertical asymptote at ( x = 3 )
- Horizontal asymptote at ( y = 1 )
For ( g(x) = \frac{2x - 1}{x + 3} ):
- Vertical asymptote at ( x = -3 )
- Horizontal asymptote at ( y = 2 )

Further Graphing Analysis

Graph additional rational functions with polynomial combinations to identify their asymptotic behavior.
Use specific example functions for practice and to visualize the asymptotic features.

Advanced Cases (Possible Examination Items)

Learn to graph rational functions where:
- ( p(x) ) is a constant and ( q(x) ) is a second-degree polynomial.
- ( p(x) ) is first-degree and ( q(x) ) is second-degree.
- ( p(x) ) is second-degree and ( q(x) ) is first-degree.

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Description

Explore the characteristics of rational functions and learn how to identify their domain and range. This quiz includes examples that require you to graph these functions using the Geogebra app. Test your understanding of the concepts and apply your skills in analyzing rational functions.