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Overview

The domain of a rational expression refers to all the possible values of the variable that make the expression defined. A rational expression is undefined when the denominator equals zero because division by zero is undefined in mathematics. Therefore, when finding the domain of a rational expression, we need to identify the values of the variable that would make the denominator equal to zero and exclude them from the domain.

Steps to Find the Domain of a Rational Expression

1. Identify the denominator(s) of the rational expression.
2. Set the denominator equal to zero.
3. Solve the equation to find the values of the variable that would make the denominator zero.
4. Exclude these values from the domain.
5. Express the domain in interval notation, excluding the values where the denominator is zero.

Example 1: Finding the Domain of a Simple Rational Expression

Consider the rational expression:

\( \frac{1}{x - 3} \)

Step 1: Identify the denominator. In this case, the denominator is \( x - 3 \).

Step 2: Set the denominator equal to zero:

\( x - 3 = 0 \)

Step 3: Solve for \( x \):

\( x = 3 \)

Step 4: Exclude \( x = 3 \) from the domain because it makes the denominator zero.

Step 5: The domain of the expression is all real numbers except \( x = 3 \). In interval notation, the domain is:

\( (-\infty, 3) \cup (3, \infty) \)

Example 2: Finding the Domain of a Rational Expression with a Quadratic Denominator

Consider the rational expression:

\( \frac{2}{x^2 - 4} \)

Step 1: Identify the denominator. The denominator is \( x^2 - 4 \).

Step 2: Set the denominator equal to zero:

\( x^2 - 4 = 0 \)

Step 3: Solve for \( x \):

\( x^2 = 4 \) → \( x = \pm 2 \)

Step 4: Exclude \( x = 2 \) and \( x = -2 \) from the domain because they make the denominator zero.

Step 5: The domain of the expression is all real numbers except \( x = 2 \) and \( x = -2 \). In interval notation, the domain is:

\( (-\infty, -2) \cup (-2, 2) \cup (2, \infty) \)

Practice Questions

1. Question 1: Find the domain of the following rational expression:

   \( \frac{3}{x + 5} \)

   Solution

   Step 1: The denominator is \( x + 5 \).

   Step 2: Set the denominator equal to zero:

   \( x + 5 = 0 \)

   Step 3: Solve for \( x \):

   \( x = -5 \)

   Step 4: Exclude \( x = -5 \) from the domain.

   Step 5: The domain is \( (-\infty, -5) \cup (-5, \infty) \).

2. Question 2: Find the domain of the following rational expression:

   \( \frac{1}{x^2 - 9} \)

   Solution

   Step 1: The denominator is \( x^2 - 9 \).

   Step 2: Set the denominator equal to zero:

   \( x^2 - 9 = 0 \)

   Step 3: Solve for \( x \):

   \( x^2 = 9 \) → \( x = \pm 3 \)

   Step 4: Exclude \( x = 3 \) and \( x = -3 \) from the domain.

   Step 5: The domain is \( (-\infty, -3) \cup (-3, 3) \cup (3, \infty) \).

3. Question 3: Find the domain of the following rational expression:

   \( \frac{5}{x^2 + 4x + 3} \)

   Solution

   Step 1: The denominator is \( x^2 + 4x + 3 \).

   Step 2: Set the denominator equal to zero:

   \( x^2 + 4x + 3 = 0 \)

   Step 3: Factor the quadratic expression:

   \( (x + 1)(x + 3) = 0 \)

   Step 4: Solve for \( x \):

   \( x = -1 \) → \( x = -3 \)

   Step 5: Exclude \( x = -1 \) and \( x = -3 \) from the domain.

   Step 6: The domain is \( (-\infty, -3) \cup (-3, -1) \cup (-1, \infty) \).

4. Question 4: Find the domain of the following rational expression:

   \( \frac{2}{x^2 - 6x + 9} \)

   Solution

   Step 1: The denominator is \( x^2 - 6x + 9 \).

   Step 2: Set the denominator equal to zero:

   \( x^2 - 6x + 9 = 0 \)

   Step 3: Factor the quadratic expression:

   \( (x - 3)^2 = 0 \)

   Step 4: Solve for \( x \):

   \( x = 3 \)

   Step 5: Exclude \( x = 3 \) from the domain.

   Step 6: The domain is \( (-\infty, 3) \cup (3, \infty) \).