The big idea: The inverse function f⁻¹ undoes what f does.
If f takes x to y, then f⁻¹ takes y back to x.
For example, if f(3) = 7, then f⁻¹(7) = 3.
- the inverse of f — reads "f inverse"
- applying f then f⁻¹ brings you back to the start
Real-world inverse: taxi fare
The total fare for a taxi ride, T dollars, for a trip of d km is modelled by
T = 2.4d + 18
Find the inverse formula giving d in terms of T.
Step by step
- Step 1 — Write the original. T is the output, d is the input.
- Step 2 — Subtract 18 from both sides.
- Step 3 — Divide by 2.4 to solve for d.
Final answer
d = (T − 18) / 2.4. This converts a fare T into the trip distance d.
Critical: f⁻¹(x) is NOT 1/f(x): The notation f⁻¹ looks like an exponent of −1, but it is NOT a reciprocal.
If f(x) = 3x − 5, then: ✅ f⁻¹(x) = (x + 5) / 3 (the inverse function) ❌ 1/f(x) = 1/(3x − 5) (the reciprocal — a completely different thing)
This is the most common confusion with inverse notation.
- inverse function f⁻¹
- The function that reverses f: if f(a) = b, then f⁻¹(b) = a.
- one-to-one function
- A function where every output comes from exactly one input — required for the inverse to also be a function.
Worked example — finding f⁻¹ from a table
The function f is defined by the table below.
| x | 0 | 1 | 2 | 3 | 4 |
|------|---|---|---|---|----|
| f(x) | 3 | 5 | 7 | 9 | 11 |
Complete the table for f⁻¹(x).
| x | 3 | 5 | 7 | 9 | 11 |
|---------|---|---|---|---|----|
| f⁻¹(x) | ? | ? | ? | ? | ? |
Step by step
- Step 1 — Remember what f⁻¹ does.
f⁻¹ undoes f. So if f(0) = 3, then f⁻¹(3) = 0.
With a table, this means swap the two rows: the inputs become outputs and vice versa. No algebra needed. - Step 2 — Read each pair backwards.
• f(0) = 3 → f⁻¹(3) = 0 • f(1) = 5 → f⁻¹(5) = 1 • f(2) = 7 → f⁻¹(7) = 2 • f(3) = 9 → f⁻¹(9) = 3 • f(4) = 11 → f⁻¹(11) = 4 - Step 3 — Fill in the table with those values.
Final answer
| x | 3 | 5 | 7 | 9 | 11 |
|---------|---|---|---|---|----|
| f⁻¹(x) | 0 | 1 | 2 | 3 | 4 |
Same pairs as f, just read backwards. No algebra — just swap the rows.
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.
How to spot it on the exam: IB will ask for f⁻¹ with a specific number inside, for example:
• Find h⁻¹(10) • Find C⁻¹(1270) • Given T⁻¹(50) = k, find k
Tackle it in 3 steps: Step 1. Set f(x) = the number from the question.
Step 2. Solve for x.
Step 3. Check by putting x back into f, then say what x means in context.
Quick example
Given
f(x) = 3x + 2
find f⁻¹(11).
Step by step
- Step 1 — Set f(x) = 11.
- Step 2 — Solve for x.
- Step 3 — Check by putting x = 3 back into f.
Final answer
f⁻¹(11) = 3.
Worked example — taxi fare
A taxi app charges T dollars for a trip of d km.
T(d) = 2.50d + 4
d ≥ 0
Find T⁻¹(24) and explain what your answer means in context.
Step by step
- Step 1 — Write the function down and say what each letter means.
• T(d) = total cost of the trip, in dollars • d = trip distance, in km - Step 2 — Find d. T⁻¹(24) asks: which d gives a fare of $24? Set T(d) = 24 and solve.
- Step 3 — Check by putting d = 8 back into T.
- Step 4 — Say what it means. Since d is km, d = 8 means a fare of $24 corresponds to an 8 km trip.
Final answer
T⁻¹(24) = 8. A trip costing $24 is 8 km long.
Worked example — rational function
A water pipe''s flow rate, F litres per minute, depends on the valve opening v mm, modelled by
F(v) = 240/v² + 0.5
3 ≤ v ≤ 12
Find F⁻¹(2.9) and interpret in context.
Step by step
- Step 1 — F⁻¹(2.9) asks: "what valve opening v gives a flow rate of 2.9?" Set up the equation.
- Step 2 — Subtract 0.5 from both sides.
- Step 3 — Multiply both sides by v², then divide by 2.4.
- Step 4 — Take the positive square root (v > 0 by context).
- Step 5 — Sense-check by substituting back.
- Step 6 — Write a sentence with context.
Final answer
F⁻¹(2.9) = 10. A valve opening of 10 mm produces a flow rate of 2.9 L/min.
You can also use the GDC for any f⁻¹(a): Same recipe — solve f(x) = a — but you let the GDC do the equation-solving. Especially useful when f involves an exponential, a logarithm, or an awkward fraction.
The walkthrough below shows it on a pipe flow-rate question.
GDC walkthrough
Step through the exact calculator keystrokes, screen by screen, in study mode.
Worked example — exponential cooling (GDC method)
A coffee cools so that its temperature, T °C, after t minutes is modelled by
T(t) = 75e^(−0.04t) + 18
t ≥ 0
Find T⁻¹(40) using the GDC, and interpret in context.
Step by step
- Step 1 — T⁻¹(40) asks: "for what t does T(t) = 40?" Set up the equation T(t) = 40.
- Step 2 — You could solve algebraically with natural log, but the GDC's intersect tool is much faster — use that.
Press Y=. Enter: - Y₁ = 75·e^(−0.04X) + 18 - Y₂ = 40
Press ZOOM → 6 (ZStandard), then GRAPH. - Step 3 — Find the intersection: 2nd → TRACE → 5: intersect. Press ENTER three times to confirm Y₁, Y₂, and a starting guess.
- Step 4 — The intersection is at t ≈ 30.7 (3 s.f.).
- Step 5 — Write a sentence with context.
Final answer
T⁻¹(40) ≈ 30.7. The coffee reaches 40 °C about 30.7 minutes after pouring.
GDC walkthrough
Step through the exact calculator keystrokes, screen by screen, in study mode.
How to interpret f⁻¹(a) = k in context: When IB asks you to interpret f⁻¹(a) in context, write a full sentence using the form:
"f⁻¹(a) = k means that an input of k gives an output of a."
Example: if T(t) is the coffee's temperature (in °C) t minutes after pouring, then T⁻¹(40) ≈ 30.7 means "the coffee reaches 40 °C about 30.7 minutes after pouring".
Always state what the input variable represents in real-world context (minutes, hours, km, mm, etc.). This sentence is usually worth its own 1 mark on the exam.
Get feedback like a real examiner
Submit your answers and get instant feedback — what you did well, what's missing, and exactly what to write to score full marks.
The three-step method: IB gives you a formula linking two real-world variables (like F = 1.8C + 32) and asks for a formula the other way.
Step 1. Start with the given formula.
Step 2. Solve for the variable IB asks for. Treat the other variable like a number — same algebra as before.
Step 3. Write the result. That's the inverse formula.
If the question uses y = f(x) notation instead of named variables, do the same: solve for x, then rename y → x so the inverse takes x as its input.
Worked example — currency conversion
A money exchange converts British pounds to US dollars using
D = 1.25P + 2
where P is the amount in pounds and D is the amount in dollars (the 2 is a fixed handling fee).
Find a formula for P in terms of D.
Step by step
- Step 1 — Start with the given formula.
- Step 2 — Solve for P. Subtract 2 from both sides.
- Divide both sides by 1.25.
- Step 3 — That's the inverse formula. No relabelling needed: P and D already have real-world names.
Final answer
P = (D − 2) / 1.25
Always write f⁻¹(x) = ...: Do not just write the expression without the correct notation.
IB awards a mark for correct notation: f⁻¹(x) = (x + 5)/3.
Writing just "(x + 5)/3" is incomplete — you haven't told IB what you found.
The big idea: Graphs of f and f-1 are reflections in the line y=x.
| Point on f | Point on f-1 |
|---|---|
| (2,7) | (7,2) |
| (0,3) | (3,0) |
For f(x) = 2x + 3 (indigo) and f⁻¹(x) = (x − 3)/2 (teal): each indigo dot has a matching teal dot with swapped coordinates. The dashed grey line y = x is the mirror.
Interactive diagram
Explore the labelled diagram, charts and maps for this topic in full study mode.
GDC walkthrough
Step through the exact calculator keystrokes, screen by screen, in study mode.
Coordinate swap: Swap x and y coordinates for corresponding points.
IB-style question — sketch the inverse [4 marks]
The diagram shows the graph of a function f drawn for the domain 0 ≤ x ≤ 6. The graph of f is a straight line from the point (0, 1) to the point (6, 4).
(a) Write down the range of f.
(b) On the same axes, sketch the graph of f⁻¹, showing the coordinates of its endpoints.
Step by step
- (a) The range of f is the set of output (y) values. As x runs from 0 to 6, the line rises from y = 1 to y = 4, so the outputs fill the interval from 1 to 4.
- (b) The graph of f⁻¹ is the reflection of f in the line y = x, so reflect each point by swapping its coordinates.
- The endpoint (0, 1) reflects to (1, 0); the endpoint (6, 4) reflects to (4, 6). Draw the straight line joining these two reflected points.
- Check: the domain of f⁻¹ equals the range of f (1 ≤ x ≤ 4) and the range of f⁻¹ equals the domain of f (0 ≤ y ≤ 6) — consistent with the reflection.
Final answer
(a) 1 ≤ y ≤ 4. (b) f⁻¹ is the line from (1, 0) to (4, 6) — the reflection of f in y = x.