Big idea (in plain English): Domain = the list of x-values you are ALLOWED to put into the function.
Ask yourself: which x-values would BREAK the function?
Throw those out.
Everything else is the domain.
| If you see… | What breaks? | Rule for the domain |
|---|---|---|
| A fraction | Bottom (denominator) cannot be 0 | Find the x that makes the bottom = 0 and EXCLUDE it |
| A square root | Inside cannot be negative | Set the inside ≥ 0 and solve |
| A normal line, parabola, etc. | Nothing breaks | Domain = all real numbers (x ∈ ℝ) |
| A real-life model (time, people, money) | Negative or silly values make no sense | Keep only values that make real-life sense |
Worked example 1 — fraction
Find the domain of f(x) = 1 / (x − 3).
Step by step
- Step 1 — Spot the danger. This is a fraction, so the denominator (x − 3) is the part that can break.
- Step 2 — Ask: what x makes the bottom = 0?
- Step 3 — That x is the only one we have to throw out.
- Step 4 — Every other real number is fine.
Final answer
Domain: x ∈ ℝ, x ≠ 3 (every real number except 3)
Worked example 2 — square root
Find the domain of g(x) = √(2x − 6).
Step by step
- Step 1 — Spot the danger. The square root means the inside cannot be negative.
- Step 2 — Set the inside to be ≥ 0.
- Step 3 — Solve like a normal inequality.
- Step 4 — Those are the allowed inputs.
Final answer
Domain: x ≥ 3
30-second exam check: Scan the function.
Is there a fraction?
A square root?
A log?
A real-life situation?
If yes → apply that rule.
If none of these → the domain is all real numbers.
Most common mistake: Forgetting to write 'x ≠ …' for fractions.
Even if the rest of the function is fine, you must exclude the value that makes the bottom 0.
Big idea: The range is all the y-values the function can produce.
Think: what can come out?
| Question | Think |
|---|---|
| Domain | What x-values can go in? |
| Range | What y-values can come out? |
Worked example — range on a restricted domain
The speed of a delivery drone, S metres per second, is measured t seconds after launch.
The speed is modelled by
S(t) = 80/t + 2
4 ≤ t ≤ 20
Find the range of S(t).
Step by step
- Step 1 — The domain is 4 ≤ t ≤ 20, so substitute both endpoints into S(t) to find the minimum and maximum outputs.
- Step 2 — Substitute the left endpoint, t = 4, to find the maximum.
- Step 3 — Substitute the right endpoint, t = 20, to find the minimum.
- Step 4 — Write the range from minimum to maximum, with units and a sentence of context.
Final answer
6 ≤ S ≤ 22. The drone's speed is between 6 m/s and 22 m/s throughout the flight.
Graph shortcut: Range is the vertical span of the graph: lowest y-value to highest y-value.
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Graph method: Domain is read left to right.
Range is read bottom to top.
| What to find | Where to look | Axis |
|---|---|---|
| Domain | How far the graph goes left and right | x-axis |
| Range | How far the graph goes down and up | y-axis |
Endpoints — read the dot: When the graph stops, look at the dot at the end.
The dot tells you whether that point is part of the graph.
| You see at the end | It means | Use |
|---|---|---|
| ● Filled dot | The point IS included | ≤ or ≥ |
| ○ Open circle | The point is NOT included | < or > |
| Arrow → | The graph keeps going forever | no upper or lower bound |
Worked example
A graph starts at a filled dot at (−2, 1), reaches (3, 8), and ends at an open circle at (6, 2).
State the domain and range.
Step by step
- Domain: read the x-values from left to right.
- The graph starts at x = −2 with a filled dot, so −2 is included.
- The graph ends at x = 6 with an open circle, so 6 is not included.
- Range: read the y-values from bottom to top.
Final answer
Domain: −2 ≤ x < 6. Range: 1 ≤ y ≤ 8.
[Diagram: math-domain-range-visualizer] - Available in full study mode
Common mistake: Do not swap them.
Domain uses x-values.
Range uses y-values.
Context changes the domain: In real-life questions, some x-values may work mathematically but make no sense in the story.
| Context | Restriction |
|---|---|
| Time after start | t ≥ 0 |
| Time during one day | 0 ≤ t ≤ 24 |
| Number of people/items | whole numbers only |
| Height above ground | height ≥ 0 |
| Percentage | 0 ≤ percentage ≤ 100 |
| One complete cycle (trig model) | 0 ≤ t ≤ period length |
Worked example — context domain
The height of a ball is h(t) = −5t² + 20t, where t is time in seconds.
State the domain in context.
Step by step
- Time cannot be negative, so t starts at 0.
- The ball lands when its height is zero. Set h(t) = 0.
- Factor: both terms share −5t, so pull it out front. (−5t × t = −5t², and −5t × (−4) = 20t, so this checks out.)
- Zero product rule: if two factors multiply to zero, at least one of them must be zero. Set each factor equal to zero on its own.
- Solve each equation.
- The ball is in the air from launch (t = 0) to landing (t = 4).
Final answer
Domain: 0 ≤ t ≤ 4 seconds
Why this works — zero product rule: If a × b = 0, then a = 0 OR b = 0 (at least one of them).
So whenever an equation becomes a product equal to zero (like −5t(t − 4) = 0), set each factor to zero on its own and solve.
This is the standard move for finding where a quadratic equals zero.
IB phrase to notice: When the question says “in context”, do not write all real numbers.
Restrict the domain to values that make sense.
The inverse trick — domain and range swap: When h has an inverse h⁻¹, their domain and range swap:
• Range of h⁻¹ = Domain of h • Domain of h⁻¹ = Range of h
You do NOT need to find the formula for h⁻¹. Just read the domain or range of h directly from the question.
Worked example — range of an inverse function
A water tank drains so that its volume, V litres, with the drain open for t minutes, is modelled by
V(t) = 500/t + 3
5 ≤ t ≤ 50
Write down the range of V⁻¹.
Step by step
- Step 1 — Identify what is being asked.
The question asks for the range of V⁻¹ (the inverse function).
The inverse trick: range of V⁻¹ = domain of V. No calculation needed. - Step 2 — Read the domain of V directly from the question.
The domain is given as 5 ≤ t ≤ 50. - Step 3 — State the range of V⁻¹. It equals the domain of V.
Final answer
Range of V⁻¹ is 5 ≤ t ≤ 50.