Question

What is the domain of a rational function?

Short answer

Question: Determine the domain of the following rational function: R(x) = \frac{4x-5}{x^2 - 16}. Answer: To find the domain of the rational function, we follow the steps outlined in the solution. 1. Identify the denominator Q(x) of the rational function: Q(x) = x^2 - 16 2. Set the denominator equal to zero (Q(x) = 0) and solve for x: x^2 - 16 = 0 (x-4)(x+4) = 0 The solutions are x = 4 and x = -4. 3. The domain of the rational function is all real numbers except the x-values that make Q(x) = 0. Therefore, the domain of the rational function R(x) = \frac{4x-5}{x^2 - 16} is all real numbers except x = 4 and x = -4.

Step-by-step solution

Defining a Rational Function

A rational function is a function that can be expressed as the quotient or fraction of two polynomials (P(x) and Q(x)) with Q(x) not equal to zero. It can be written as: R(x) = \frac{P(x)}{Q(x)}

Understanding Domain

The domain of a function is the set of all possible input values (x-values) for which the function is defined. In other words, it's the set of x-values for which the function produces a valid output or value.

Determining the Domain of a Rational Function

To determine the domain of a rational function, we need to find all x-values that make the denominator (Q(x)) equal to zero, since division by zero is undefined in mathematics. 1. Identify the denominator Q(x) of the rational function. 2. Set the denominator equal to zero (Q(x) = 0) and solve for x. 3. The domain of the rational function is all real numbers except the x-values that make Q(x) = 0.

Example

Let's find the domain of a sample rational function: R(x) = \frac{2x + 3}{x^2 -9} The denominator of the rational function is Q(x) = x^2 - 9. Now we set Q(x) equal to zero and solve for x: x^2 - 9 = 0 (x-3)(x+3) = 0 The solutions are x = 3 and x = -3. Therefore, the domain of the rational function R(x) is all real numbers except x = 3 and x = -3.