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.”
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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.
GDC walkthrough
Step through the exact calculator keystrokes, screen by screen, in study mode.
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.
Drag the vertical line across each graph. If it ever touches two points at once, it fails the test.
Interactive diagram
Explore the labelled diagram, charts and maps for this topic 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.