Square root function domain calculator
√(x - 4) is defined only for x ≥ 4, so its domain is [4, ∞) and the graph starts at 4 on the x-axis. Enter a and b for √(ax + b) and get the domain in interval notation. Free in your browser.
- Free
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- No sign-up
- Your data stays on your device
Run — free
Worked example
- A
- 1
- B
- -4
- Function
- sqrt(x-4)
- Radicand
- x-4
- Inequality
- x-4 ≥ 0
- Solution
- x ≥ 4
Finding the domain of a square root function is one of the first interval-notation skills in algebra and precalculus. For a real-valued expression under a radical, the radicand must be greater than or equal to zero; otherwise the square root is not defined over the reals. This square root function domain calculator accepts the linear coefficients a and b in f(x) equals the square root of ax plus b, solves the inequality ax plus b greater than or equal to zero, and returns the domain in interval notation together with the starting point of the graph when a is nonzero. Positive a produces a left-closed ray starting at negative b over a; negative a produces a right-closed ray ending at that same critical value; constant nonnegative radicands yield all real numbers.
Why the radicand must be nonnegative
In the real numbers the square root symbol denotes the principal nonnegative root of a nonnegative quantity. That definition fails as soon as the expression inside the radical dips below zero: there is no real number whose square is negative. For the linear family f(x) equals the square root of ax plus b, the domain condition is therefore the single linear inequality ax plus b greater than or equal to zero.
Solving that inequality is pure algebra. When a is positive you divide without flipping the inequality and obtain x greater than or equal to negative b over a, so the domain is the closed unbounded interval from that starting point to positive infinity. The boundary point is always included because equality is allowed: the square root of zero is zero, so the graph of y equals the square root of ax plus b begins (or ends) on the x-axis at the starting point and never continues into the forbidden half-line.
Where finding square root domains helps
Classrooms and tutoring platforms use a square root function domain calculator so learners can check interval notation for arbitrary linear radicands without flipping inequality signs by hand when a is negative. Exam-prep generators need a deterministic oracle that marks empty constant radicands as invalid instead of silently returning an empty array that a buggy grader might treat as success. Failed validation does not consume balance.
Keep requests small, pin the rounded starting point and interval string in tests, and you get reproducible domains for homework, LMS plugins, and edge workers without standing up symbolic math infrastructure of your own. When your curriculum later introduces nested radicals or quadratic radicands, treat this tool as the linear foundation: same nonnegative rule, different inequality to solve.
What you can do with it
Classroom and homework checks
Students submit a and b for f(x)=√(ax+b) and verify the ordered interval and starting point against textbook methods.
Notebooks and precalculus pipelines
Gate real square-root steps on ax+b≥0 in scripts without importing a full computer-algebra library.
FAQ
What happens when a is zero and b is negative?
The radicand is a negative constant for every x, so the real domain is empty.
How do I find the domain of a square root function?
Require the expression under the root to be zero or more and solve that inequality. For √(x - 4) that means x - 4 ≥ 0, so x ≥ 4 and the domain is [4, ∞).
Why is the starting point included in the domain?
Because the square root of zero is zero, so equality is allowed. The graph begins exactly on the x-axis at that point, which is why interval notation uses a square bracket on the finite side.
What changes when a is negative?
Dividing by a negative number flips the inequality. With a of -2 and b of 8 you get x ≤ 4, so the domain is (-∞, 4] and the graph ends at 4 instead of starting there.
What if a is zero and b is zero or positive?
Then the expression under the root is a constant that is never negative, so the function is defined for every real number and the domain is the whole real line.
Related tools
For developers — API access
Everything on this page is available programmatically. This section is for teams who want to wire it into their own systems; everyone else can just use the tool above.
API endpoint
Prefer to automate it? One authenticated POST creates the task; the result comes back by webhook or a signed link. The same capability also runs here on the web, by email and from Telegram — and soon from our app too.
Call it from your stack
curl -X POST https://api.kit.forhosting.com/algebra/square-root-function-domain \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"a":1,"b":-4}'const res = await fetch("https://api.kit.forhosting.com/algebra/square-root-function-domain", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"a": 1,
"b": -4
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/algebra/square-root-function-domain",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"a": 1,
"b": -4
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/algebra/square-root-function-domain", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"a":1,"b":-4}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"a":1,"b":-4}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/algebra/square-root-function-domain", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"a": 1,
"b": -4
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "algebra.square_root_function_domain",
"status": "queued",
"_links": {
"result": "/tasks/tsk_…/result"
}
}The API is asynchronous: the call returns a task_id immediately and the result arrives by webhook. Polling is capped at 1 req/s per task.
Pricing
Published price — no tokens, no invented credits. A failed task is never charged.
Errors
| HTTP | Code | Meaning |
|---|---|---|
401 | unauthorized | Missing or invalid API key. |
402 | insufficient_balance | Your balance doesn't cover the task price. |
404 | unknown_type | That task type doesn't exist. |
429 | rate_limited | Too many requests. Use the webhook instead of polling. |
Technical notes
Empty domains when the radicand is a negative constant are rejected with invalid_input instead of a hollow success payload.
Why the radicand must be nonnegative
When a is negative you flip the inequality and obtain x less than or equal to the same critical value, so the domain is negative infinity through that endpoint inclusive. When a is zero the radicand collapses to the constant b: if b is nonnegative the square root is defined for every real x and the domain is the entire real line; if b is negative there is no real x at all and this calculator reports invalid_input rather than inventing an empty interval that some graders might misread as success. That closed-endpoint habit matches every standard textbook and is the reason interval notation uses a square bracket on the finite side of the domain.
Input contract, response shape, and error behaviour
Example with a negative slope inside: a equals negative two and b equals eight yields x less than or equal to four and domain left-paren negative infinity comma four right-bracket. When a is zero and b is negative the task fails with a clear empty-domain message and is not billed. POST a JSON body with two required finite numbers: a, the coefficient of x inside the radicand, and b, the constant term. Each field may arrive as a JSON number or as a numeric string; missing fields, empty strings, and non-finite values fail with invalid_input naming the field. On success the body echoes rounded a and b, a function string such as sqrt(x-4), a radicand string, the inequality restating ax plus b greater than or equal to zero, a solution string for x, the domain in interval notation with infinity symbols, starting_point equal to negative b over a whenever a is nonzero, bound_side describing whether the finite endpoint is on the left or right of the interval (or none when the domain is all reals), starting_inclusive true when a finite endpoint is present, empty_domain false on every success path, and a short formula field you can show next to the answer. Example: a equals one and b equals negative four yields f(x) equals the square root of x minus four, solution x greater than or equal to four, domain left-bracket four comma infinity right-paren, and starting_point four. Pair the numbers with your own graphing code if you need a sketch of the half-parabola; this endpoint only solves the domain inequality and does not emit plots or multi-step prose beyond the formula field.
Where a square root domain API earns its keep
Engineering notebooks that gate sensors or physical models behind a real square root can call a tiny parse API instead of pulling a full computer-algebra system for a two-coefficient inequality.
Integration use cases
Exam and LMS auto-grading
Mark domain-of-radical items with a deterministic oracle that rejects negative constant radicands and never invents open endpoints.
Technical questions
How do I find the domain of √(ax + b)?
Send finite numbers a and b. The API solves ax+b≥0 and returns interval notation plus starting_point=−b/a when a≠0 (closed at that endpoint).
Why is the endpoint included in the domain?
Equality is allowed: √0 = 0 is real. Interval notation therefore uses a square bracket at the finite starting point.
What if a is negative?
Dividing by a negative flips the inequality, so the domain is (−∞, −b/a] with starting_point still equal to −b/a on the right.
What does the square root function domain calculator cost?
The browser tool is free. API requests use prepaid KIT balance at $0.002 per successful call; invalid input is not billed.