Problem 87 - College Algebra Essentials

Question

Explain how to determine which numbers must be excluded from the domain of a rational expression.

Step-by-Step Solution

Answer

To determine which numbers must be excluded from the domain of a rational expression, set the denominator equal to zero and solve the equation. The x-values found are the ones that must be excluded from the domain, as they would make the denominator equal to zero, rendering the rational expression undefined. For example, in the expression \( \frac{x+2}{x-1} \), 1 is excluded from the domain.

Step 1: Identify the Rational Expression

A rational expression is a fraction in which the numerator and/or the denominator are polynomials. An example might look like this: \( \frac{x+2}{x-1} \). In order to determine the domain, we need to identify what values for x would make the denominator zero, as this would make the expression undefined.

Step 2: Set Denominator Equal to Zero

Next, take the denominator of your rational expression and set it equal to zero. For our example expression, this would look like: \( x - 1 = 0 \).

Step 3: Solving the Equation

Solve the equation to find out the value of x. For our example this would be \( x = 1 \). This step gives us the x-value that would make the denominator of our rational expression zero.

Step 4: Determining Excluded Numbers

The number obtained is the one that should be excluded from the domain, so for our example, 1 would be excluded from the domain as it would make the denominator zero. That means any real number except 1 is included in the domain.