Inputs in, outputs out: The domain is the set of all x-values you're allowed to put in.
The range is the set of all y-values that come out.
Think of the machine: Domain = what you can feed the machine; range = what it can produce.
If nothing stops you, the domain is all real numbers (x ∈ ℝ).
IB-style question — domain and range of x²
State the domain and range of f(x) = x².
Step by step
- Any real number can be squared — nothing is banned.
- A square is never negative, and every value ≥ 0 is reachable.
Final answer
Domain x ∈ ℝ; range y ≥ 0.
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Left–right for domain, down–up for range: Read the domain off the x-axis — how far the graph spreads left to right.
Read the range off the y-axis — how far it spreads down to up.
Step through it: read left→right for the domain, check whether each end is filled (included) or open (excluded), then read bottom→top for the range.
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IB-style question — read off a parabola
The graph of y = x² − 4 is a parabola with lowest point (0, −4).
State its domain and range.
Step by step
- The parabola extends forever left and right.
- Its lowest output is −4 and it opens upward.
Final answer
Domain x ∈ ℝ; range y ≥ −4.
y = x² − 4 runs left–right forever (so domain = ℝ), but never dips below its lowest point −4 — the outputs fill only the shaded band, giving range y ≥ −4.
Interactive diagram
Explore the labelled diagram, charts and maps for this topic in full study mode.
Open vs closed ends: A filled dot (or solid endpoint) includes that value — use ≤ or ≥.
An open dot excludes it — use < or >.
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Two bans: ÷0 and √(negative): A formula works for every x except where it would divide by zero or square-root a negative.
So set any denominator ≠ 0, and keep anything under an even root ≥ 0.
(A logarithm needs its argument > 0.)
IB-style question — a denominator
Find the domain of f(x) = 1/(x − 3).
Step by step
- The denominator can't be zero.
- Solve.
Final answer
Domain: all real x except x = 3 (x ≠ 3).
IB-style question — a square root
Find the domain of g(x) = √(x − 2).
Step by step
- What's under the root can't be negative.
- Solve.
Final answer
Domain: x ≥ 2.
√0 is fine — but not in a denominator: √0 = 0 is allowed, so use ≥.
But if that root sits in a denominator, it must be > 0 (it can't be zero and can't be negative).