--- title: "Domain and Range of a Function" description: "Describe the set of inputs a function accepts and the set of outputs it can return, and say what forces a restriction: a division, a square root, or the situation being modelled." canonical: https://lightmysky.com/learn/mathematics/domain-and-range-of-a-function-mt_b_4zoHz8Jc source: https://lightmysky.com/learn/mathematics/domain-and-range-of-a-function-mt_b_4zoHz8Jc.md retrieved: 2026-09-02 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Domain and Range of a Function Describe the set of inputs a function accepts and the set of outputs it can return, and say what forces a restriction: a division, a square root, or the situation being modelled. Subject: Mathematics · Area: Algebra · Ages 16 to 17 Page: https://lightmysky.com/learn/mathematics/domain-and-range-of-a-function-mt_b_4zoHz8Jc ## Ready when they can - State the domain of f(x) = 1/(x - 3) and say why 3 is left out - Read the range of a quadratic from its turning point - Give a sensible domain for a function whose input is a length ## Lesson: Domain and Range of a Function Last stop you sketched three curves and saw where each one refuses to go. The reciprocal never touches the axes. The exponential stays above the x-axis however far left you look. This stop gives that idea its two names. The Ridgeway recreation ground is being rebuilt this season, and every rule the crew writes accepts some inputs and refuses others. The set of inputs a function accepts is called its domain. The set of outputs that come back is called its range. A rule can be sound algebra and still have inputs it cannot take. *(drawing: The domain is the gate at the front. The range is what comes out of the back.)* The first thing that cuts inputs out is a division. Dividing by zero has no answer, so any input that makes a denominator zero is refused. Take f(x) = 1/(x - 3). Set the denominator to zero: x - 3 = 0, so x = 3. Every other input goes through, so the domain is every real number apart from 3. The rest of the rule is sound. One input has no output, and on the graph it shows up as the vertical line the curve runs beside and never reaches. The second thing is a square root. No real number squares to a negative, so the inside of a square root has to be zero or more. For f(x) = √(x - 2) the inside gives x - 2 >= 0, which solves to x >= 2. That is the domain. The range takes a second question. The square root sign returns the root that is not negative, so the smallest output is 0, reached at x = 2. The range is y >= 0. A rule can carry both restrictions at once, and then an input has to clear every one of them. *(drawing: Trying inputs one at a time shows you where the domain starts.)* The third restriction comes from the situation, not from the algebra. The bowl holds 500 litres and the pump takes out 20 litres a minute, so V(t) = 500 - 20t. The algebra is happy with t = -3 or t = 40. The pump is not. Time starts at 0, and the bowl is empty when 500 - 20t = 0, so t = 25. The sensible domain is 0 <= t <= 25, and the range is 0 <= V <= 500. You get that by asking what the input stands for. **Example.** A quadratic accepts every input, so its domain is the whole number line. Its range is the interesting half, and the turning point hands it to you. Take h(x) = x² - 6x + 11. Complete the square: h(x) = (x - 3)² + 2. A square is never negative, so the smallest value of (x - 3)² is 0, reached at x = 3. That makes 2 the lowest output, and every value above 2 is reached as well. The range is h(x) >= 2. If the x² term were negative, the curve would turn over and 2 would be the largest output instead. **Recap.** The domain is the set of inputs a rule accepts and the range is the set of outputs it gives back. A division bars the input that makes the denominator zero, a square root bars every input that makes the inside negative, and a real situation bars whatever makes no sense on site. Read a quadratic's range from its turning point. ## Practice 24 questions on this page, each with its working shown. ## Needs first - [Recognising Cubic, Reciprocal and Exponential Graphs](https://lightmysky.com/learn/mathematics/recognising-cubic-reciprocal-and-exponential-graphs-mt_2IJJ51rmq9) - [Function Notation and Evaluating Functions](https://lightmysky.com/learn/mathematics/function-notation-and-evaluating-functions-mt_YWKWzHK_6Z) - [Inverse Functions](https://lightmysky.com/learn/mathematics/inverse-functions-mt_7k7q5ZjMb4) ## Opens up - [Polynomial Division and the Factor Theorem](https://lightmysky.com/learn/mathematics/polynomial-division-and-the-factor-theorem-mt_haL2UkGa5C)