Find the domain and range of each function in Exercises...

Find the domain and range of each function in Exercises $27-32 .$ Use a graphing utility or plot points to sketch a graph of the function, and illustrate the domain and range on the graph.
$f(x)=\frac{1}{x^{2}-1}$

Key Concepts

Graphing Techniques

Graphing techniques involve plotting points or using a graphing utility to visualize the function’s behavior. This includes marking any discontinuities, such as asymptotes, and clearly illustrating the domain and range on the graph. This approach helps reinforce the understanding of how the function behaves across all its valid input and output values.

Rational Functions

Rational functions are functions that can be expressed as a ratio of two polynomials. Key aspects of rational functions include identifying points of discontinuity where the denominator becomes zero, as well as understanding how the degrees of the polynomials in the numerator and denominator affect the overall behavior of the function.

Asymptotes

Asymptotes are lines that a graph of a function approaches but never actually reaches. For rational functions, vertical asymptotes occur where the denominator is zero (and the numerator is nonzero), and horizontal or slant asymptotes describe the end-behavior of the function as the independent variable grows large in magnitude. They are essential in understanding the limits and behavior of the function both near points of discontinuity and at infinity.

Range

The range of a function is the set of all possible output values the function can produce. It involves analyzing the behavior of the function, such as its asymptotic behavior and potential restrictions on outputs due to the function’s formula or inherent limits, which is often illustrated by looking at its graph.

Domain

The domain of a function represents all the input values for which the function is defined. It involves determining which values can be substituted into the function without causing any mathematical issues, such as division by zero or taking the square root of a negative number. In the context of rational functions, this typically means excluding any values that make the denominator equal to zero.

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