The big idea: A function is a rule. Every input gives exactly one answer.__LINEBREAK__What matters is that there is always one answer. Not two. Not zero. One.
Example: the rule f(x) = 2x + 1 says "double the input, then add 1".__LINEBREAK__Put in 3 → double it → 6 → add 1 → 7.__LINEBREAK__So f(3) = 7. The output is 7.
Real-world anchor: A currency converter is a function. Type in 10 USD → you get one answer: €9.20. Not sometimes €9 and sometimes €10. Always one answer for the same input.__LINEBREAK__A lottery number is NOT a function of your ticket — the same ticket could "give" different numbers on different days.
- function
- A rule where every input has exactly one output. Same input → same output, every time.
- input (x)
- The value you put into the function. Also called the argument.
- output f(x)
- The value the rule produces for that input. Also called the image or function value.
| 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. One input, two outputs. |
| y = x² | Yes ✓ | Each x gives exactly one y-value. |
| x² + y² = 9 (a circle) | No ✗ | x = 0 gives y = 3 or y = −3. Two outputs for one input. |
IB exam: "Is this a function? Justify.": Always give a reason — the mark is for the justification, not just the yes/no.__LINEBREAK___✅ Full answer: "No, because when x = 4 there are two possible outputs (+2 and −2), so each input does not give exactly one output."__LINEBREAK___❌ Bare answer: "No." — likely no mark.
The big idea: f(x) is read "f of x". It means: the output of function f when the input is x.__LINEBREAK__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
Two functions at once
Given f(x) = x + 2 and g(x) = x², find f(5) and g(3).
Step by step
- Find f(5): replace x with 5 in f.
- Find g(3): replace x with 3 in g.
Final answer
f(5) = 7 and g(3) = 9. Always work on each function separately.
Critical trap: f(2) does NOT mean f × 2.__LINEBREAK__f(2) means "the output when x = 2". It is a substitution, not a multiplication.__LINEBREAK__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.__LINEBREAK__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.__LINEBREAK__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__LINEBREAK___−3² = −9 (the square only applies to 3)__LINEBREAK__−3² = 9 (the square applies to negative 3)__LINEBREAK__Always write brackets around negative numbers before squaring.
Evaluating with an expression as input
Find f(a + 1) when f(x) = 2x − 3.
Step by step
- Replace every x with (a + 1). Use brackets.
- Expand the bracket.
- Simplify.
Final answer
f(a + 1) = 2a − 1. This is an expression, not a number.
Show the substitution step: Always write the line where you substitute before simplifying.__LINEBREAK__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.__LINEBREAK__If the ruler ever touches the graph in two places at the same time → the graph is NOT a function.__LINEBREAK__If the ruler always touches the graph in exactly one place (or skips it entirely) → the graph IS a function.__LINEBREAK__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."__LINEBREAK___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."__LINEBREAK___❌ 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.__LINEBREAK__Example: the right half of x² + y² = 9 is a function, because no vertical line hits it twice.