Domain of Rational Function Calculator
Find the domain of rational functions in interval and set-builder notation
Enter polynomial (use ^ for exponents)
Enter polynomial expression
Understanding Domain of Rational Functions
What is Domain?
The domain of a function is the set of all possible input values (x-values) for which the function is defined. For rational functions, we must exclude any values that make the denominator zero.
Finding the Domain
- Set the denominator equal to zero: denominator = 0
- Solve for x to find excluded values
- The domain is all real numbers except these excluded values
- Express the domain in interval or set-builder notation
Notation Types
Example: (-∞, 2) ∪ (2, ∞) means all real numbers except 2
Example: {x ∈ ℝ | x ≠ 2} means "all real numbers x such that x is not 2"
Example
Find domain of f(x) = (x + 1)/(x² - 4)
- Set denominator = 0: x² - 4 = 0
- Factor: (x + 2)(x - 2) = 0
- Solve: x = -2 or x = 2
- Domain (interval): (-∞, -2) ∪ (-2, 2) ∪ (2, ∞)
- Domain (set-builder): {x ∈ ℝ | x ≠ -2, x ≠ 2}
Graph Interpretation
Excluded values correspond to vertical asymptotes on the graph. The function approaches ±∞ as it gets close to these values but never crosses them.
Frequently Asked Questions
Why can't the denominator be zero?
Division by zero is undefined in mathematics. When the denominator equals zero, the function has no value at that point, creating a vertical asymptote on the graph.
Does the numerator affect the domain?
No! Only the denominator determines the domain. The numerator can be zero (making the function value 0), but the denominator cannot.
What's the difference between () and [] in interval notation?
Parentheses ( ) mean the endpoint is NOT included (open interval). Brackets [ ] mean the endpoint IS included (closed interval). For rational functions, we always use ( ) for excluded values.
Can a rational function have no restrictions?
Yes! If the denominator is a non-zero constant, the domain is all real numbers. Example: f(x) = x/5 has domain (-∞, ∞).
What if the denominator is always positive?
If the denominator can never equal zero (like x² + 1), the domain is all real numbers. The denominator being positive or negative doesn't matter - it just can't be zero.
How do I write 'all real numbers' in notation?
Interval notation: (-∞, ∞). Set-builder notation: {x ∈ ℝ} or {x | x is a real number}. Both mean the same thing.
What's a vertical asymptote?
A vertical asymptote is a vertical line (x = a) where the function approaches infinity. It occurs at domain restrictions where the denominator equals zero.
Can I simplify before finding the domain?
NO! Always find the domain from the original function. If you simplify first and cancel factors, you'll miss restrictions. Example: (x-2)/(x-2) simplifies to 1, but x ≠ 2 is still a restriction.