How can I find the domain of a function from its formula?
Quick answer
To find a formula’s domain over the real numbers, start with all real inputs and rule out any that make the formula undefined or nonreal. For beginner examples, check that every denominator is nonzero and that the expression inside each square root is at least zero. Solve those conditions, combine them, and write the allowed inputs in interval or set notation. [S1]
The question, the details, the next step.
What should I check first when a function is given as a formula?
Assume the question is asking for the real-number domain unless it says otherwise. The domain is the set of input values that make the formula produce a defined real output. Start by looking for parts that can make an output undefined or nonreal. In beginner formulas, the two checks to make are denominators and square roots: a denominator cannot be zero, and the expression under a real square root must be at least zero. [S1]
A plain polynomial such as f(x) = 2x + 5 has no denominator or square root to restrict its inputs, so its domain is all real numbers, written (-∞, ∞). [S1] For other formulas, inspect each restriction rather than assuming every real number works.
How do I turn a denominator into a restriction?
Set the denominator equal to zero to find the input that must be excluded. Then state that all other real inputs are allowed, unless another part of the formula adds a restriction.
Worked example: f(x) = 1/(x - 3).
- The denominator is x - 3. It would be zero when x - 3 = 0, so x = 3 is not allowed.
- The domain is all real x except 3: {x ∈ ℝ | x ≠ 3}.
- In interval notation, that is (-∞, 3) ∪ (3, ∞).
The parentheses mean 3 is left out. The union sign joins the allowed values on either side of 3.
How does the check change when there is a square root?
For a real-valued square root, set the expression inside the root greater than or equal to zero, then solve that inequality. The boundary value is included when it makes the expression inside the root zero. [S1]
Worked example: g(x) = √(x + 2).
- Require x + 2 ≥ 0.
- Solve to get x ≥ -2.
- The domain is [-2, ∞), or {x ∈ ℝ | x ≥ -2}.
The square bracket includes -2 because √0 is defined as a real number. This rule is for real-valued square roots; other kinds of functions may have different domain restrictions.
Can you show how to combine the checks and give me a quick routine to reuse?
Apply every restriction, then keep only the inputs that satisfy all of them. For example, with h(x) = √(x + 2)/(x - 1), the root requires x ≥ -2, while the denominator requires x ≠ 1. Combine those conditions: x ≥ -2 and x ≠ 1. The domain is [-2, 1) ∪ (1, ∞), or {x ∈ ℝ | x ≥ -2 and x ≠ 1}. [S1]
Reusable routine:
- State that you are working over the real numbers.
- Find each denominator and require it to be nonzero.
- Find each square-root expression and require it to be at least zero.
- Solve each condition and combine them; an input is allowed only if it passes every check.
- Write the result in interval or set notation and check whether endpoints are included or excluded.
This routine covers the denominator and real square-root restrictions shown here. Other formula types may require additional checks.
A practical checklist
Assume real-number inputs unless the problem specifies a different setting.
For each denominator, exclude values that make it zero.
For each real square root, require the expression inside to be at least zero.
Combine conditions so every retained input satisfies all restrictions.
Use brackets for included endpoints, parentheses for excluded endpoints, and ∪ to join intervals.
A few more useful answers
Do I include an input that makes the expression inside a square root equal to zero?
Yes, for a real square root: zero is allowed, so include that boundary if the other restrictions permit it. [S1]
What if an input is allowed by the square-root condition but makes a denominator zero?
Exclude it. Every condition must hold at once; one failed condition makes the input unavailable.
Is a domain always written in interval notation?
No. You can use interval notation or set notation, as long as the notation clearly shows which inputs are allowed.
Sources
References used for this guide.
- [S1] OpenStax College Algebra with Corequisite Support 2e: Domain and Range openstax.org · Checked