Graph each function. Give the domain and range. See Examples...

Key Concepts

Graphical Asymptotes

Asymptotes are lines that the graph of a function approaches but never actually reaches. Vertical asymptotes typically occur at values that make the denominator zero in a rational function, while horizontal asymptotes describe the behavior of the function as the input grows large in magnitude. Grasping these concepts is essential for accurately graphing rational functions.

Range of Functions

The range of a function is the set of all output values it can produce. For rational functions, analyzing the reach of the function often involves understanding the limits as the input variable approaches values that result in asymptotic behavior, and determining the output values that the function cannot attain.

Domain of Functions

The domain of a function is the set of all possible input values for which the function is defined. In the case of rational functions, you must exclude any values that make the denominator zero, as these lead to undefined expressions.

Rational Functions

Rational functions are functions defined by a ratio of two polynomials. They exhibit important features such as discontinuities where the denominator equals zero, and different asymptotic behaviors depending on the degrees of the numerator and denominator. Understanding their structure is crucial for determining their graphs and analyzing their properties.

Transcript

Please give Ace some feedback