In Exercises 76–81, find the domain of each function. f(x) = x/(x^2 + 4x -21)

Verified step by step guidance

1. Identify the function type: The function is a rational function, which is a fraction where both the numerator and the denominator are polynomials.
2. Determine the domain: The domain of a rational function is all real numbers except where the denominator is zero.
3. Set the denominator equal to zero: Solve the equation x^2 + 4x - 21 = 0 to find the values of x that make the denominator zero.
4. Factor the quadratic equation: Factor x^2 + 4x - 21 into (x + 7)(x - 3) = 0.
5. Solve for x: Set each factor equal to zero, x + 7 = 0 and x - 3 = 0, to find the values of x that are not in the domain.

Key Concepts

Domain of a Function

The domain of a function refers to the set of all possible input values (x-values) for which the function is defined. For rational functions, the domain is typically restricted by values that make the denominator zero, as division by zero is undefined.

Finding Roots of a Quadratic Equation

To determine the values that restrict the domain, one must find the roots of the quadratic equation in the denominator. This involves using methods such as factoring, completing the square, or applying the quadratic formula to identify the x-values that make the denominator zero.

Inequalities and Exclusions

Once the roots of the denominator are found, these values must be excluded from the domain. The domain can then be expressed in interval notation, indicating all real numbers except for the points where the function is undefined due to division by zero.