Find The Domain of The Following Rational Function Calculator

What is the Domain of a Rational Function?

The domain of a rational function is the set of all real numbers for which the function is defined. For a rational function in the form:

where P(x) and Q(x) are polynomials, the domain includes all real numbers except those that make the denominator Q(x) equal to zero.

In other words, the domain of a rational function is all real numbers except the values of x that make the denominator zero.

How to Find the Domain of a Rational Function

To find the domain of a rational function, follow these steps:

1. Identify the denominator of the rational function.
2. Set the denominator equal to zero and solve for x.
3. Exclude any x-values that make the denominator zero from the domain.
4. Express the domain in interval notation.

Examples of Finding Domains

Example 1

Find the domain of f(x) = (x + 3)/(x - 2).

Solution:

1. The denominator is x - 2.
2. Set x - 2 = 0 → x = 2.
3. Exclude x = 2 from the domain.
4. The domain is all real numbers except 2, written as (-∞, 2) ∪ (2, ∞).

Example 2

Find the domain of f(x) = (x² - 9)/(x² - 4).

Solution:

1. The denominator is x² - 4.
2. Set x² - 4 = 0 → x = ±2.
3. Exclude x = 2 and x = -2 from the domain.
4. The domain is all real numbers except -2 and 2, written as (-∞, -2) ∪ (-2, 2) ∪ (2, ∞).

Common Mistakes to Avoid

When finding the domain of a rational function, it's easy to make these common mistakes:

• Forgetting to exclude values that make the denominator zero.
• Including values that make the numerator zero (this is correct, but the denominator must not be zero).
• Making errors when solving the equation Q(x) = 0.
• Incorrectly writing the domain in interval notation.

Double-check your work to avoid these mistakes.