For Exercises 99-102, a) Determine the domain of the...

Key Concepts

Horizontal Asymptotes

Horizontal asymptotes describe the end behavior of a rational function as x approaches positive or negative infinity. They are determined by comparing the degrees of the numerator and denominator polynomials, which helps in understanding the long-term behavior of the function.

Vertical Asymptotes

Vertical asymptotes occur in rational functions at the values of x that make the denominator zero (provided those zeros are not canceled by corresponding factors in the numerator). They represent values where the function grows without bound and are crucial for understanding the function's behavior near discontinuities.

Graphing Techniques for Rational Functions

Graphing a rational function involves plotting points, identifying intercepts, and drawing asymptotes to accurately depict the behavior of the function. Connected graphing mode emphasizes the continuity of the function across intervals while clearly representing discontinuities at asymptotes.

Rational Functions

A rational function is defined as a fraction where both the numerator and the denominator are polynomials. Understanding rational functions is essential because their properties, such as discontinuities and asymptotic behavior, are determined by the factors of the numerator and denominator.

Domain of a Function

The domain of a function is the set of all possible input values for which the function is defined. For rational functions, the domain must exclude values that make the denominator zero, as division by zero is undefined.

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