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.
Free preview
This is the free notes preview
You're reading the free notes. Aimnova Pro unlocks the full study experience — and you can try it with your first topic free to keep:
- FlashcardsLock in vocabulary and key terms with spaced repetition.
- Practice questionsAnswer exam-style questions and get instant AI marking.
- Mock exams & past-paper vaultSit full mocks and see exactly how examiners award marks.
- Personalised study planA daily plan built around your exam date and weak areas.
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.
Know your predicted grade
Take timed mock exams and get detailed feedback on every answer. See exactly where you're losing marks.
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.
Step 1: read the graph left to right for the domain. Step 2: check whether endpoints are filled or open. Step 3: read bottom to top for the range.
Interactive diagram
Explore the labelled diagram, charts and maps for this topic 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.