The big idea: The domain is the set of all valid input values (x-values) for a function.__LINEBREAK__Think of it as: what can you put in? Some values are forbidden — for example, you cannot divide by zero or square-root a negative number.
- domain
- The complete set of valid x-values (inputs) for a function.
- restriction
- A value that must be excluded from the domain because it causes a mathematical problem (e.g. division by zero, square root of a negative).
Domain of a rational function
Find the domain of f(x) = 1/(x − 3).
Step by step
- Ask: what value of x causes a problem?
- x = 3 makes the denominator zero — undefined.
- State the domain: all real numbers except x = 3.
Final answer
Domain: all real numbers except x = 3.
Domain of a square root function
Find the domain of g(x) = √(2x − 6).
Step by step
- You cannot square root a negative number. Set the expression inside ≥ 0.
- Solve.
- State the domain.
Final answer
Domain: x ≥ 3.
| Function type | Restriction rule | Domain example |
|---|---|---|
| f(x) = 1/(x−a) | denominator ≠ 0 → x ≠ a | x ∈ ℝ, x ≠ a |
| f(x) = √(x−a) | inside square root ≥ 0 → x ≥ a | x ≥ a |
| f(x) = log(x−a) | argument > 0 → x > a | x > a |
| f(x) = 2x + 5 | no restriction | all real numbers |
IB notation for domain: IB accepts inequality notation: x ≥ 3, or x ∈ ℝ, x ≠ 3.__LINEBREAK__For an integer-only domain (e.g. number of cars), write: x ∈ ℤ, x ≥ 0.__LINEBREAK__If the context limits the values (e.g. "t is time in hours from 0 to 24"), always state the domain as 0 ≤ t ≤ 24.
The big idea: The range is the set of all output values (y-values) that the function actually produces.__LINEBREAK__While the domain asks "what can go in?", the range asks "what can come out?"__LINEBREAK__Note: the range is NOT all of ℝ just because the domain is. Some outputs may be impossible.
Range of a quadratic
Find the range of f(x) = x² + 1 for all real x.
Step by step
- x² is always ≥ 0 for any real x.
- So x² + 1 is always ≥ 1.
- The minimum output is 1 (when x = 0). The function can grow without limit.
Final answer
Range: f(x) ≥ 1. The graph never goes below y = 1.
Range of an exponential
Find the range of g(x) = 2ˣ for all real x.
Step by step
- 2ˣ is always positive — it never equals zero or goes negative.
- As x → −∞, 2ˣ → 0 but never reaches 0.
- As x → +∞, 2ˣ grows without bound.
Final answer
Range: g(x) > 0 (all positive real numbers).
[Diagram: math-domain-range-visualizer] - Available in full study mode
| Domain | Range | |
|---|---|---|
| What it is | All valid x-values (inputs) | All possible y-values (outputs) |
| Axis | Horizontal (x-axis) | Vertical (y-axis) |
| On a graph | The left-to-right span | The bottom-to-top span |
| Notation | x ≥ 2, or 0 < x < 5 | y ≥ 0, or 1 < y < 10 |
Range from a graph: Look at the vertical extent of the graph.__LINEBREAK__If the graph goes from y = 2 up to y = 7 (and reaches both endpoints), the range is 2 ≤ y ≤ 7.__LINEBREAK__Filled dots = endpoints included. Empty circles = endpoints excluded.
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The big idea: Domain from a graph = the horizontal span — how far left and right does the graph reach?__LINEBREAK___Range from a graph = the vertical span — how far up and down does the graph reach?__LINEBREAK__Always check the endpoint markers: filled dot (●) means included, open circle (○) means excluded.
Reading from a graph
A graph of f(x) starts at a filled dot at (−2, 1), reaches a maximum at (3, 8), and ends at an open circle at (6, 2). State the domain and range.
Step by step
- Domain: the x-values run from −2 (included) to 6 (excluded).
- Range: the y-values run from 1 (the lowest point, included) to 8 (the maximum, included).
Final answer
Domain: −2 ≤ x < 6. Range: 1 ≤ y ≤ 8.
Wrong
- Domain: 1 ≤ x ≤ 8 (reading the y-span instead of x-span)
- Range: −2 ≤ y < 6 (reading the x-span instead of y-span)
- Ignoring the open circle — writing ≤ instead of <
Correct
- Domain: look at the x-axis span, left to right
- Range: look at the y-axis span, bottom to top
- Match the bracket type to the dot type: filled = ≤, open = <
[Diagram: math-domain-range-visualizer] - Available in full study mode
IB mark allocation: IB usually awards one separate mark for domain and one for range.__LINEBREAK__Write each on a new line, clearly labelled.__LINEBREAK___Domain: −2 ≤ x < 6 Range: 1 ≤ y ≤ 8__LINEBREAK__Using "f(x)" instead of "y" for range is also accepted: 1 ≤ f(x) ≤ 8.
The big idea: In real-world models, the domain is restricted because some x-values make no physical sense.__LINEBREAK__For example: time cannot be negative. Number of cars cannot be a fraction. Temperature cannot exceed a physical limit.__LINEBREAK__IB always says "in the context of this model" — this is your signal to restrict the domain.
Context-restricted domain
The height of a ball above the ground is modelled by h(t) = −5t² + 20t, where t is time in seconds. State the domain of h in context.
Step by step
- First ask: when does the ball leave the ground and when does it land?
- Solve: the ball is on the ground at t = 0 (launch) and t = 4 (landing).
- The ball is in the air between these times. Negative t has no meaning.
Final answer
Domain: 0 ≤ t ≤ 4 seconds. (The ball is above ground only during this interval.)
| Context | Variable | Typical domain restriction |
|---|---|---|
| Time after start | t (hours/seconds) | t ≥ 0 |
| Number of items | n (cars, people, etc.) | n ∈ ℤ, n ≥ 0 |
| Temperature model over a day | t (hours) | 0 ≤ t ≤ 24 |
| Height above ground | h (metres) | h ≥ 0 |
| Percentage | P (%) | 0 ≤ P ≤ 100 |
What IB awards the mark for: IB awards the mark for a domain answer that:__LINEBREAK__1. Uses correct inequality notation 2. Makes sense in the real-world context 3. Matches the endpoints that are physically meaningful__LINEBREAK__Writing "x ≥ 0" when the question involves time is correct. Writing "all real numbers" ignores the context and loses the mark.
Shortcut for context domain questions: Read the question for clues: "t is the number of hours after midnight" → 0 ≤ t ≤ 24. "n is the number of full years" → n ∈ ℤ, n ≥ 0.__LINEBREAK__If unsure, state the domain as the interval where the model makes physical sense and explain briefly.