A function is a machine: A function f is a rule: put a number in, get exactly one number out.
f(x) is the output for input x — so f(3) means "put 3 into the rule."
IB-style question — read the machine
For f(x) = 2x + 1, find f(3) and f(−2).
Step by step
- f(3): replace every x with 3.
- f(−2): replace every x with −2 (use brackets).
Final answer
f(3) = 7 and f(−2) = −3.
f(x) is NOT f times x: f(x) is read "f of x" — the function applied to x.
The brackets hold the input, they are not multiplication.
One input, one output: For something to be a function, each input may give only one output.
(Different inputs can share an output — that's allowed.)
Drag the vertical line across each graph. If it ever meets the curve at two points at once, that single input has two outputs — so it is not a function. A parabola passes; a sideways curve fails.
Interactive diagram
Explore the labelled diagram, charts and maps for this topic in full study mode.
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 free for 7 days:
- 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.
Replace every x, then simplify: To find f(a), write a in place of every x — wrapping it in brackets so signs and powers behave — then simplify.
IB-style question — a negative input
For g(x) = x² − 4x, find g(−3).
Step by step
- Substitute x = −3 in brackets.
- Square and multiply.
- Add.
Final answer
g(−3) = 21.
IB-style question — an algebraic input
For f(x) = 3x − 5, find f(2a).
Step by step
- Replace x with the whole expression 2a.
- Simplify.
Final answer
f(2a) = 6a − 5.
Brackets save you: Without brackets, (−3)² becomes −9 by mistake.
Always write (−3)²= 9.
The same care applies when the input is an expression.
Study smarter, not longer
Most students waste 40% of study time on topics they already know. Our AI tracks your progress and optimizes every minute.
f(a) reads UP then ACROSS; f(x) = k reads ACROSS then DOWN: Given a graph, to find f(a) go up from a on the x-axis to the curve, then across to the y-axis.
To solve f(x) = k, go across from y = k to the curve, then down — and watch for more than one answer.
Click 'Walk me through it' for the read-off step by step, then switch to 'Practice' and read a few off yourself before revealing.
Interactive diagram
Explore the labelled diagram, charts and maps for this topic in full study mode.
IB-style question — read from the graph
The graph of y = f(x) for −1 ≤ x ≤ 5 is shown above.
Write down:
(a) f(2);
(b) f(0);
(c) the values of x for which f(x) = 5.
Step by step
- (a) f(2): go UP from x = 2 to the curve, then ACROSS to the y-axis.
- (b) f(0): up from x = 0, then across.
- (c) f(x) = 5: go ACROSS from y = 5 to the curve, then DOWN — it meets the curve twice.
Final answer
(a) 6 (b) 2 (c) x = 1 and x = 3.
f(x) = k can have more than one answer: Reading across often hits the curve twice — give all the x-values.
And don't mix them up: f(a) starts on the x-axis (read up); solving f(x) = k starts on the y-axis (read across).
Given the output, find the input: Evaluating goes input → output.
Solving f(x) = k goes the other way: set the rule equal to k and solve for x.
IB-style question — a linear rule
For f(x) = 2x + 1, solve f(x) = 9.
Step by step
- Set the rule equal to 9.
- Solve.
Final answer
x = 4.
IB-style question — two inputs, one output
For f(x) = x² − 3, solve f(x) = 6.
Step by step
- Set equal to 6.
- Solve — remember both roots.
Final answer
x = 3 or x = −3 (two inputs give the same output).
Don't lose a solution: Quadratics (and other curves) can send two inputs to the same output, so f(x) = k may have more than one answer — write them all.
Practice with real exam questions
Answer exam-style questions and get AI feedback that shows you exactly what examiners want to see in a full-marks response.
A table is just f written out: Each column pairs an input x with its output f(x).
To read f(a), find a in the x-row and take the value directly below it.
To solve f(x) = k, scan the f(x)-row for k and read off the x above it.
A second row (like g) works exactly the same way.
| x | −1 | 0 | 2 | 5 |
|---|---|---|---|---|
| f(x) | 4 | 2 | −1 | 6 |
| g(x) | 2 | 5 | 0 | −1 |
IB-style question — from a table
The table above shows values of f(x) and g(x); both f and g are one-to-one.
Find:
(a) g(0);
(b) f(2);
(c) the value of x for which f(x) = 6.
Step by step
- (a) Read g(0) straight from the g-row, under x = 0.
- (b) Read f(2) from the f-row, under x = 2.
- (c) Scan the f-row for 6 and read the x above it.
Final answer
(a) 5 (b) −1 (c) x = 5.
Down for a value, across for an input: To read f(a) or g(a), go down the column under a.
To solve f(x) = k, go across the f-row to find k, then read the x above it.
The two questions move in different directions — don't mix them up.