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

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Key Concepts

Rational Functions

Rational functions are functions expressed as the quotient of two polynomials. They are characterized by possible discontinuities where the denominator equals zero, which often leads to vertical asymptotes. Understanding these functions involves analyzing their behavior over different intervals, especially near these discontinuities.

Domain

The domain of a function is the complete set of possible input values (x-values) for which the function is defined. For a rational function, this involves excluding any x-value that makes the denominator zero, as division by zero is undefined.

Range

The range of a function consists of all possible output values (y-values). For rational functions, the range may be limited by horizontal or slant asymptotes or other constraints imposed by the function’s structure, meaning certain y-values may never be attained.

Graphing Functions

Graphing a function involves plotting points that satisfy the function and identifying key features such as intercepts, asymptotes, and overall behavior. For rational functions, special attention is paid to the locations of vertical asymptotes (where the function is undefined) and horizontal asymptotes (which indicate the behavior of the function as x approaches infinity or negative infinity).

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