Domain and Range: Meaning, Examples, and Practice Problems

The Common Core State Standards for high school functions describe this as a core relationship: a function assigns to every element of its domain exactly one element of its range, and f(x) denotes the output that corresponds to a given input x. Keeping that one-to-one relationship in mind makes it easier to apply the right rule from the table above.

How to Find Domain and Range

Finding domain and range comes down to two separate checks: what can safely go in, and what can actually come out. Watching a worked video example alongside these steps helps the pattern stick faster than reading alone.

Steps to find the domain

1. Identify the function type: polynomial, rational, radical, and so on.
2. If it is a polynomial, the domain is all real numbers, since there are no restrictions.
3. If it is a rational function, set the denominator equal to zero and exclude those x-values.
4. If it is a square root function, set the expression under the root greater than or equal to zero and solve.
5. If a specific interval is given with the function, use that interval directly as the domain.
6. Write the final answer in interval notation, or as “all real numbers except” a listed set of values.

Steps to find the range

1. Write the function as y = f(x), then solve for x in terms of y, producing x = g(y).
2. Find the domain of this new function x = g(y). That domain is the range of the original function.
3. For quadratics, locate the vertex instead: the range starts or ends at its y-value, depending on which way the parabola opens.
4. For graphs, trace the curve from its lowest to highest y-value rather than solving algebraically.

Domain and Range Examples

Working through real problems is the fastest way to see these rules in action. Here are worked examples covering the function types students run into most often.

Example 1. Rational function

Find the domain and range of f(x) = 1/(x − 2).

The denominator cannot equal zero, so x ≠ 2. The domain is (−∞, 2) ∪ (2, ∞). Since this function’s output can never actually reach zero, the range is (−∞, 0) ∪ (0, ∞).

Example 2. Square root function

Find the domain and range of g(x) = √(x + 5).

The expression under the root must be non-negative, so x + 5 ≥ 0, giving x ≥ −5. The domain is [−5, ∞), and since a square root never produces a negative result, the range is [0, ∞).

Example 3. Quadratic function

Find the domain and range of h(x) = −2(x − 1)² + 7.

Quadratic functions accept every real number, so the domain is (−∞, ∞). The vertex is at (1, 7), and since the coefficient is negative, the parabola opens downward. The range is (−∞, 7].

Example 4. Absolute value function

Find the domain and range of k(x) = |x + 3| − 4.

Absolute value expressions accept any real number, so the domain is (−∞, ∞). The smallest output happens when the expression inside the bars equals zero, giving k(x) = −4, so the range is [−4, ∞).

How a Domain and Range Calculator Works

A domain and range calculator takes a function’s equation as input and returns its domain and range instantly, usually alongside a graph. Most calculators scan the equation for the same restrictions covered above: zero denominators, negative radicands, and undefined logarithmic inputs, then express the result in interval notation.

These tools help you check your work quickly, especially on longer rational or piecewise functions where one arithmetic slip changes the entire answer. But relying on a calculator without understanding why a restriction exists tends to backfire on exams, where you need to show the reasoning, not just state the final interval. The better habit is to solve the problem by hand first, then verify the result.

Think10x.ai’s AI explainer video tool turns a domain and range question, typed, spoken, or photographed from a worksheet, into a video that shows every step of the reasoning.

Domain and Range Practice Problems (Worksheet)

Use the domain and range problems listed below to test what you have learned. Work through each problem before checking the solution underneath it. If you get stuck, a narrated video walkthrough can show the full reasoning instead of just a static answer key.

Problem 1. Find the domain of f(x) = (x + 1)/(x² − 9).

Solution

Set x² − 9 = 0, which factors to (x − 3)(x + 3) = 0, giving x = 3 and x = −3. The domain is (−∞, −3) ∪ (−3, 3) ∪ (3, ∞).

Problem 2. Find the domain and range of g(x) = 3√(2x − 4).

Solution

Set 2x − 4 ≥ 0, giving x ≥ 2. The domain is [2, ∞). Since the square root output is always non-negative, the range is [0, ∞).

Problem 3. Find the range of h(x) = x² + 6x + 5.

Solution

The vertex x-coordinate is −b/2a = −3. Substituting gives h(−3) = 9 − 18 + 5 = −4. Since the parabola opens upward, the range is [−4, ∞).

Checklist before you submit an answer

• Did you check for zero denominators in every rational expression?
• Did you check for negative values under every square root?
• Did you write the final answer in interval notation, not just a description?
• Did you double-check the range using the vertex or a quick substitution, not just the domain rules?

How Think10x. ai Helps You Master Domain and Range

Instead of a static list of steps, Think10x.ai generates a narrated, animated video for the exact problem a student is stuck on, whether it comes from a typed question, a worksheet photo, or a spoken query. Students can pause mid-explanation and ask a follow-up question, such as why a denominator gets excluded, and the AI tutor picks up exactly where the video left off.

In partnership with Vidyamandir Classes (VMC), Think10x.ai’s video explanations helped clear over 80% of student doubts independently, reaching more than 8,500 students without a teacher re-explaining each concept one-on-one. 

Teachers building practice sets can use the platform’s AI math solver to generate a full walkthrough for any function type in seconds, and students working independently get the same depth of explanation available on demand through the Think10x.ai homepage.

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