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.
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.
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.
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) |
[Diagram: math-inverse-reflection] - Available in full 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.