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

Domain of a Function

This concept refers to the complete set of all possible input values (typically 'x' values) that a function can accept without causing any mathematical issues, such as division by zero, taking the square root of a negative number, or logarithms of non-positive values. Determining the domain involves identifying any restrictions from these operations to ensure the function is properly defined.

Range of a Function

The range of a function is the set of all possible output values (typically 'y' values) that the function can produce when it operates over its domain. Analyzing the range often involves understanding the behavior of the function at its extreme values and its asymptotic tendencies, which helps in identifying the minimum and maximum outputs that are attainable or the limits the function approaches.

Rational Functions

Rational functions are expressed as the quotient of two polynomials. They have distinct characteristics, such as potential restrictions in the domain due to undefined points caused by a zero denominator. Additionally, they can exhibit various asymptotic behaviors and continuity properties, which must be examined to fully understand how they behave over their domain and range.

Graphing Functions

Graphing functions is a method of visually representing the relationship between variables. This approach is valuable for identifying key features of the function, such as its domain, range, intercepts, and asymptotes. Tools like graphing utilities or plotting points by hand are commonly used to cross-check analytical findings and gain a deeper understanding of the function's behavior.

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