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.
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.
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.
Open vs closed ends: A filled dot (or solid endpoint) includes that value — use ≤ or ≥. An open dot excludes it — use < or >.
Feeling unprepared for exams?
Get a clear study plan, practice with real questions, and know exactly where you stand before exam day. No more guessing.
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).
The range follows the shape: Read the range from the graph's shape: a quadratic turns at its vertex (a minimum or maximum), and an exponential aˣ stays positive.
IB-style question — a quadratic's range
State the range of f(x) = (x − 2)² + 3.
Step by step
- Vertex form a(x − h)² + k: the vertex is (2, 3) and it opens up.
- So the smallest output is 3.
Final answer
Range: y ≥ 3.
IB-style question — an exponential's range
State the range of f(x) = 2ˣ.
Step by step
- 2ˣ is always positive and gets close to 0 but never reaches it.
Final answer
Range: y > 0.
Vertex gives the boundary: For a quadratic in vertex form a(x − h)² + k: the range is y ≥ k (opens up, a > 0) or y ≤ k (opens down, a < 0).