For Exercises 63 through 76, identify each function type....

For Exercises 63 through $76,$ identify each function type. Then find the domain of each function. Write the domain using both interval notation and inequality notation.
$$g(x)=\frac{2 x+9}{x^{2}+4 x-5}$$

Key Concepts

Inequality Notation

Inequality notation provides an alternative method to express the domain of a function by describing the set of allowable values using inequalities. This method delineates the range of valid inputs by specifying the conditions each value must satisfy.

Interval Notation

Interval notation is a concise way to describe the set of all real numbers that form the domain of a function. It uses parentheses to denote exclusion of endpoints and brackets to indicate inclusion, clearly showing continuous ranges of allowed values.

Rational Functions

A rational function is defined as the quotient of two polynomials, where the numerator and denominator are both polynomial expressions. The key aspect of rational functions is that they can be undefined for certain values that make the denominator equal to zero, which in turn restricts the domain of the function.

Domain of a Function

The domain of a function encompasses all the values for which the function is defined. For rational functions, the domain must exclude any values that result in an undefined operation, such as division by zero, ensuring the function gives valid outputs.

Denominator Restrictions

When working with rational functions, the denominator plays a crucial role in determining the domain. The values that make the denominator zero lead to undefined expressions (division by zero) and must therefore be omitted from the domain.

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