The big idea: A function is a rule where every input gives exactly one output.
Same input → same answer, every time.
Not two answers. Not random answers. Just one.
Example: the rule f(x) = 2x + 1 means “double the input, then add 1”.
Put in 3 → 2(3) + 1 = 7
So f(3) = 7.
Real-world anchor: Think of a vending machine.
Press B4 → you should ALWAYS get the same snack.
Same button → same result.
That is a function.
If pressing B4 sometimes gave chips, sometimes chocolate, and sometimes a drink, it would NOT be a function because one input would give multiple outputs.
- function
- A rule where every input has exactly one output.
- input (x)
- The value you put into the function.
- output f(x)
- The answer the function gives.
| Relation | Is it a function? | Why? |
|---|---|---|
| x → x + 2 | Yes ✓ | Every x gives exactly one answer. |
| x → ±√x | No ✗ | x = 4 gives two answers: +2 and −2. |
| y = x² | Yes ✓ | Each x gives one y-value. |
| x² + y² = 9 | No ✗ | x = 0 gives y = 3 or y = −3. |
IB exam tip: Always explain WHY.
✅ “No, because x = 4 gives two outputs (+2 and −2), so one input gives more than one output.”
❌ “No.”
The big idea: f(x) is read "f of x". It means: the output of function f when the input is x.
The letter f is just a name — you can also see g(x), h(x), or P(t) in IB questions. They all work the same way.
- the name of the function
- the input (goes inside the bracket)
- the rule — what happens to x
Reading function notation
If f(x) = 3x − 5, find f(4).
Step by step
- The input is 4. Replace every x with 4.
- Simplify.
Final answer
f(4) = 7
Worked example — bike rental cost
A bike rental shop charges C(d) = 45d + 25 dollars for a rental of d days.
Find C(4) and say what your answer means.
Step by step
- Step 1 — Re-read the question in plain words.
d is the number of days. C(d) is the total cost in dollars.
So C(4) is really asking: "how much will the shop charge me for a 4-day rental?"
To answer it, put 4 wherever you see d in the formula. - Step 2 — Put 4 in place of d.
- Step 3 — Multiply first (× before +). 45 × 4 = 180.
- Step 4 — Add.
- Step 5 — Answer the original question.
The shop will charge $205 for a 4-day rental.
Final answer
C(4) = 205. A 4-day rental costs $205.
Critical trap: f(2) does NOT mean f × 2.
f(2) means "the output when x = 2". It is a substitution, not a multiplication.
If f(x) = 3x − 5, then f(2) = 3(2) − 5 = 1, not 5 × 2 = 10.
Always write the function value line: Write f(4) = 3(4) − 5 before simplifying.
IB awards a mark for the correct substitution line — even if your arithmetic goes wrong after that.
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The big idea: To evaluate a function at a value: replace every x in the rule with that value.
Then simplify the result. Always bracket negative numbers when substituting.
Evaluating a quadratic
Find f(−3) when f(x) = x² + 2x − 1.
Step by step
- Replace every x with (−3). Use brackets.
- Calculate each term.
- Simplify.
Final answer
f(−3) = 2
Bracket negative inputs every time: −3² ≠ (−3)²
−3² = −9 (the square only applies to 3)
(−3)² = 9 (the square applies to negative 3)
Always write brackets around negative numbers before squaring.
Worked example — soup cooling
Mira poured a bowl of soup and let it cool.
The soup's temperature, S, in °C, t minutes later is modelled by:
S(t) = 60e^(−0.05t) + 22
where t ≥ 0.
Find the soup's temperature 12 minutes after it was poured.
Step by step
- Step 1 — Re-read the question in plain words.
t is the time in minutes after the soup was poured. S(t) is the soup's temperature in °C.
So S(12) is really asking: "how hot is the soup 12 minutes after Mira poured it?"
To answer it, put 12 wherever you see t in the formula. - Step 2 — Substitute t = 12.
- Step 3 — Tidy the exponent first. −0.05 × 12 = −0.6.
- Step 4 — Type into your GDC and round to 3 significant figures (IB's default precision).
- Step 5 — Answer the original question.
About 12 minutes after pouring, the soup is approximately 54.9 °C.
Final answer
S(12) ≈ 54.9 °C — the soup is about 54.9 °C twelve minutes after pouring.
The same procedure works for any function type: Linear, exponential, rational, log, sinusoidal — they all use the same idea: substitute → simplify → interpret.
For example, if h(t) = 20/(2t + 5), then h(0.5) = 20/(2(0.5) + 5) = 20/6 ≈ 3.33.
Worked example — function defined by a table
A function f is given by the following table:
| x | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| f(x) | 5 | 2 | 4 | 1 | 3 |
(a) Find f(2).
(b) Solve f(x) = 4.
Step by step
- Part (a) — Find f(2).
Look up x = 2 in the top row, then read the value below: f(2) = 4. - Part (b) — Solve f(x) = 4.
Read the table backwards: find the column where f(x) = 4.
The bottom row shows 4 under x = 2. So x = 2. - Sense-check.
We found f(2) = 4 in part (a) — so solving f(x) = 4 gives x = 2. The two answers match because they ask the same relationship from opposite directions.
Final answer
(a) f(2) = 4. (b) x = 2.
Show the substitution step: Always write the line where you substitute before simplifying.
IB gives credit for showing f(3) = 2(3) − 3 even if the final simplification is wrong.
The big idea: Imagine holding a ruler upright and slowly sliding it across a graph from left to right.
If the ruler ever touches the graph in two places at the same time → the graph is NOT a function.
If the ruler always touches the graph in exactly one place (or skips it entirely) → the graph IS a function.
This is called the vertical line test.
Why does this work?
A function must give one output per input.
Each position of the ruler is one x-value (one input).
If the ruler hits two points, that x-value produces two outputs — which breaks the rule.
[Diagram: math-vertical-line-test] - Available in full study mode
| Graph | Vertical line hits… | Is it a function? |
|---|---|---|
| Straight line y = 2x + 1 | One point — every time | ✅ Function |
| Parabola y = x² | One point — every time | ✅ Function |
| Horizontal line y = 4 | One point — every time | ✅ Function |
| Circle x² + y² = 9 | Two points (top and bottom) | ❌ Not a function |
| Sideways parabola x = y² | Two points (above and below) | ❌ Not a function |
| Vertical line x = 3 | Infinite points | ❌ Not a function |
Exam answer template: If it IS a function: "Yes — every vertical line crosses the graph at most once, so each x-value gives exactly one y-value."
If it is NOT a function: "No — a vertical line at x = [value] crosses the graph at two points, so that x-value gives two different y-values."
❌ Never just write "Yes" or "No" alone — IB always awards a mark for the justification.
Restricted graph? Test only what is drawn: If IB shows only part of a curve (e.g. only the right half of a circle), apply the test only to the part that is drawn — not the full curve.
Example: the right half of x² + y² = 9 is a function, because no vertical line hits it twice.