What Domain and Range Mean
The domain is the set of inputs a function accepts. The range is the set of outputs it can produce. For f(x) = √x you can feed in 0 or any positive number, so the domain is [0, ∞), and every output is 0 or above, so the range is [0, ∞) too.
Interval notation uses a square bracket when an endpoint is included and a round bracket when it is not. Infinity always gets a round bracket because you can never reach it.
The Three Restrictions That Limit a Domain
| Trouble spot | Rule | Example |
|---|---|---|
| Denominator | Cannot be zero | 1/(x − 5): x ≠ 5 |
| Even root | Inside must be ≥ 0 | √(x + 4): x ≥ −4 |
| Logarithm | Inside must be > 0 | ln(5 − x): x < 5 |
f(x) = √(x + 4)/(x − 2)
- Square root: x + 4 ≥ 0, so x ≥ −4.
- Denominator: x − 2 ≠ 0, so x ≠ 2.
- Combine the two conditions.
- Start with [−4, ∞) and remove the point 2.
Answer: [−4, 2) ∪ (2, ∞)
How to Find the Range of a Function
The range is usually harder than the domain. Three ideas cover most cases: use the vertex for a parabola, use asymptotes for a rational function, and use the domain restriction for roots and logs.
f(x) = −x² + 4x + 1
- The graph is a parabola opening downwards because the x² coefficient is negative.
- Find the vertex x-value: x = −b/(2a) = −4/(2·(−1)) = 2.
- Find its height: f(2) = −4 + 8 + 1 = 5.
- A downward parabola never goes above its vertex.
Answer: (−∞, 5]
f(x) = (2x + 1)/(x − 3)
- The degrees of top and bottom are equal, so there is a horizontal asymptote at y = 2 (ratio of leading coefficients).
- The graph approaches 2 but never reaches it.
- Every other value is reached, which you can confirm by solving y = (2x + 1)/(x − 3) for x: x = (3y + 1)/(y − 2), which exists whenever y ≠ 2.
Answer: (−∞, 2) ∪ (2, ∞)
Domain of a Logarithmic Function
f(x) = ln(5 − x)
- A logarithm needs a positive input: 5 − x > 0.
- Solve: x < 5, so the domain is (−∞, 5).
- As x approaches 5 from the left, 5 − x approaches 0 and the log falls without limit.
- As x goes to −∞, 5 − x grows and the log grows without limit. Every real value is reached.
Answer: Domain (−∞, 5), range (−∞, ∞)
Three Restrictions Cover Most Problems
| If you see | Rule | Example | Domain |
|---|---|---|---|
| A fraction | denominator ≠ 0 | 1 ÷ (x − 5) | x ≠ 5 |
| An even root | inside ≥ 0 | √(x − 2) | [2, ∞) |
| A log | argument > 0 | ln(x + 1) | (−1, ∞) |
Combine them with “and”. For √(x − 2) ÷ (x − 5) you need x ≥ 2 and x ≠ 5, so the domain is [2, 5) ∪ (5, ∞).
Our Take: What Is Overrated, Underrated and Plain Wrong
Overrated: finding range by algebra alone. Sketch the graph or find the vertex and end behaviour first.
Underrated: reading domain left to right and range bottom to top on a graph. It is the quickest way to see both.
Plain wrong: mixing them up. Domain is the set of allowed inputs (x). Range is the set of outputs (y).
Common Functions at a Glance and Mistakes
| Function | Domain | Range |
|---|---|---|
| x² | (−∞, ∞) | [0, ∞) |
| √x | [0, ∞) | [0, ∞) |
| 1/x | x ≠ 0 | y ≠ 0 |
| eˣ | (−∞, ∞) | (0, ∞) |
| ln x | (0, ∞) | (−∞, ∞) |
| sin x | (−∞, ∞) | [−1, 1] |
- Using a square bracket next to infinity.
- Reporting x ≠ 2 but forgetting the square-root condition, or the other way round.
- Guessing the range from a few points instead of using the vertex or asymptote.
Why Students Get Stuck on Domain and Range, and What a Good Calculator Should Do
Searching for help with domain and range usually starts late at night with one problem that will not work out. Restrictions hide in denominators, roots and logs, and the range is harder to see. A good calculator fixes that by showing each step, checking its own answer and explaining the idea in plain words. This section covers what students struggle with, what they want, and how this page responds.
In short: students need to list every restriction and read the range from the graph, not just see a final number, and a calculator earns trust only when its steps can be checked.
Understand the pain points
What students get stuck on
Most mistakes here are not about arithmetic. They come from one rule applied the wrong way round, which is why seeing every step matters.
- Forgetting that a denominator cannot be zeroDomain of sqrt(x - 3)↗
- Missing that a square root needs a non-negative insideDomain of 1/(x^2 - 4)↗
- Confusing interval notation brackets and parenthesesDomain of ln(x + 2)↗
- Guessing the range without a graphRange of x^2 + 1↗
Understand the frustrations and desires
What they feel, and what they want
Each frustration points to a wish: a clearer step, a built-in check, an explanation in plain words.
- Wants each restriction named and solved
- Wants the answer in interval and set notation
- Wants a graph to confirm the range
Build the page around their intent
Each intent has a place on this page
The page follows intent, not features, so each need has a clear starting point.
- I need an answer now: type the problem into the calculator at the top of this page.Domain of sqrt(x - 3)↗
- I need to understand: the sections below explain the idea in plain words and list the common mistakes.
- I need to check my work: every step is shown, so the first line that differs from yours is the one to study.
- I need practice: open the solver, tap + and choose Create Practice Test.
- I need it explained aloud: in the solver, tap + and choose Create Video.