In each of Exercises 5-10, match the function described with...

In each of Exercises $5-10,$ match the function described with the appropriate domain from those listed below. a) $\{x | x \neq 3\}$ b) $\{x | x \neq-3\}$ c) $\{x | x \neq 5\}$ d) $\{x | x \neq-2, x \neq 5\}$ e) $\{x | x \neq 2\}$ f) $\{x | x \neq 2, x \neq 5\}$ $$ g(x)=\frac{x+3}{(x+2)(x-5)} $$

In each of Exercises $5-10,$ match the function described with the appropriate domain from those listed below.
a) $\{x | x \neq 3\}$
b) $\{x | x \neq-3\}$
c) $\{x | x \neq 5\}$
d) $\{x | x \neq-2, x \neq 5\}$
e) $\{x | x \neq 2\}$
f) $\{x | x \neq 2, x \neq 5\}$
$$
g(x)=\frac{x+3}{(x+2)(x-5)}
$$

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Key Concepts

Rational Functions

A rational function is defined as the quotient of two polynomials. Its domain is typically restricted to those values that do not make the denominator equal to zero, as division by zero is undefined. This concept is central in understanding the behavior of rational functions and determining where the function is valid.

Excluded Values from the Denominator

When working with rational functions, the values that make the denominator equal to zero are excluded from the domain. Identifying these values involves setting each factor of the denominator equal to zero and solving for the variable. This process ensures that the function's domain only includes numbers that keep the function well-defined.

Domain of a Function

This concept refers to the set of all possible input values (typically real numbers) for which a function is defined. Determining a function's domain is essential because it tells us which values can be used without leading to undefined expressions, such as division by zero or negative values under even roots, among other restrictions.

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