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Questions and Answers
Which of the following represents a rational function?
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.
The function $f(x) = \frac{1}{x}$ has a domain that includes 0.
False (B)
What is the domain of the function $g(x) = \frac{1}{x + 1}$?
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 __________.
The range of the function $f(x) = \frac{x}{x - 1}$ is __________.
Match the following functions with their corresponding domain:
Match the following functions with their corresponding domain:
What is the common feature of the graphs of the functions $f(x)$ and $g(x)$?
What is the common feature of the graphs of the functions $f(x)$ and $g(x)$?
Identify the range of the function $f(x) = \frac{1}{x}$.
Identify the range of the function $f(x) = \frac{1}{x}$.
For the function $G(x) = \frac{2x}{x + 3}$, the value that x cannot take is __________.
For the function $G(x) = \frac{2x}{x + 3}$, the value that x cannot take is __________.
What is the vertical asymptote of the function $g(x) = \frac{2x - 1}{x + 3}$?
What is the vertical asymptote of the function $g(x) = \frac{2x - 1}{x + 3}$?
The horizontal asymptote of the function $g(x) = \frac{2x - 1}{x + 3}$ is y = 2.
The horizontal asymptote of the function $g(x) = \frac{2x - 1}{x + 3}$ is y = 2.
Identify one point on the graph of the function $g(x) = \frac{2x - 1}{x + 3}$ when x = -5.
Identify one point on the graph of the function $g(x) = \frac{2x - 1}{x + 3}$ when x = -5.
The equation of the horizontal asymptote is __________.
The equation of the horizontal asymptote is __________.
Match the following functions with their descriptions:
Match the following functions with their descriptions:
What is the domain of the function defined by $f(x) = \frac{x - 2}{x - 3}$?
What is the domain of the function defined by $f(x) = \frac{x - 2}{x - 3}$?
The range of the function $g(x) = \frac{2x - 1}{x + 3}$ includes the value 2.
The range of the function $g(x) = \frac{2x - 1}{x + 3}$ includes the value 2.
Identify the vertical asymptote for the function $g(x) = \frac{2x - 1}{x + 3}$.
Identify the vertical asymptote for the function $g(x) = \frac{2x - 1}{x + 3}$.
The vertical asymptote of the function $f(x) = \frac{x - 2}{x - 3}$ is __________.
The vertical asymptote of the function $f(x) = \frac{x - 2}{x - 3}$ is __________.
Match the following asymptotes with their corresponding functions:
Match the following asymptotes with their corresponding functions:
What does the range of the function $f(x) = \frac{x - 2}{x - 3}$ exclude?
What does the range of the function $f(x) = \frac{x - 2}{x - 3}$ exclude?
The graph of $g(x) = \frac{2x - 1}{x + 3}$ will intersect the line y = 2.
The graph of $g(x) = \frac{2x - 1}{x + 3}$ will intersect the line y = 2.
For the function $g(x) = \frac{2x - 1}{x + 3}$, what is the horizontal asymptote?
For the function $g(x) = \frac{2x - 1}{x + 3}$, what is the horizontal asymptote?
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.
