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: temperature conversion
The formula F = 1.8C + 32 converts Celsius (C) to Fahrenheit (F). Find the inverse formula that converts Fahrenheit to Celsius.
Step by step
- Write the original function. F is the output, C is the input.
- Solve for C (the original input) in terms of F (the original output).
- Divide by 1.8.
Final answer
The inverse is C = (F − 32) / 1.8. This is exactly the type of inverse question IB asks.
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.
The three-step method: 1. Write y = f(x) 2. Swap x and y (swap the variable roles) 3. Solve for y
The result y = ... is f⁻¹(x) — rename it properly.
Finding the inverse of a linear function
Find f⁻¹(x) for f(x) = 3x − 5.
Step by step
- Write y = f(x).
- Swap x and y.
- Solve for y: add 5 to both sides.
- Divide by 3.
- Write the inverse using correct notation.
Final answer
f⁻¹(x) = (x + 5) / 3
Finding the inverse of a fractional function
Find f⁻¹(x) for f(x) = (x + 2) / 4.
Step by step
- Write y = f(x).
- Swap x and y.
- Multiply both sides by 4.
- Solve for y.
- Write using inverse notation.
Final answer
f⁻¹(x) = 4x − 2
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.
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What does f⁻¹(a) actually mean?: Imagine f(x) is a machine: you put a number in, the machine spits a different number out.
f⁻¹(a) asks the reverse question: "what did I put IN, to get a OUT?"
So if f takes 5 and spits out 17, then f⁻¹(17) = 5. Same machine, just read backwards.
Big consequence: to find f⁻¹(a), you don't need the inverse formula at all. Just answer the question "which x gives f(x) = a?" — that x is your answer.
How to find f⁻¹(a) in 3 steps: Step 1 — Set up. Write the equation f(x) = a (use the formula for f, put a on the right).
Step 2 — Solve for x. Undo whatever f did, in reverse order (last operation first).
Step 3 — Check. Put your answer back into f. If you get a, you're right. Then write a sentence saying what x means in context.
Worked example — taxi fare
A taxi app charges T(d) = 2.50d + 4 dollars for a trip of d km, where d ≥ 0. Find T⁻¹(24) and explain what your answer means in context.
Step by step
- Step 1 — Translate the notation into a question we can solve. T⁻¹(24) means "what distance d gives a total fare of $24?" So we need to solve T(d) = 24.
- Step 2 — Undo T's operations in reverse order. T does two things to d: multiplies by 2.50, then adds 4. To reverse, undo the +4 first, then the ×2.50. (Like taking off shoes then socks — reverse the order you put them on.)
- Step 2a — Subtract 4 from both sides.
- Step 2b — Divide both sides by 2.50.
- Step 3 — Sense-check. Put d = 8 back into the original T to confirm.
- Step 4 — Write a sentence with context. d represents km, so 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.
Common mistake — reversing in the wrong order: When you undo the operations, you must go backwards from the order they were applied.
❌ Wrong: divide by 2.50 first → d = 24/2.50 − 4 ≈ 5.6 (gives the wrong answer)
✅ Right: subtract 4 first (because +4 was the LAST thing T did), then divide by 2.50.
Rule of thumb: last-in, first-out — the operation done LAST is the one you undo FIRST.
Skip deriving the whole formula: You could find the full inverse formula T⁻¹(d) = (d − 4)/2.50 and then substitute 24. Same answer.
But for ONE input value, that's extra work for nothing. When IB asks for f⁻¹(a single number), solve f(x) = a directly. Always write the one-sentence interpretation if the question is in context — it is usually worth its own mark.
The big idea: For an inverse to exist as a function, the original function must be one-to-one — every output comes from exactly one input.
A parabola like f(x) = x² is NOT one-to-one: f(3) = 9 and f(−3) = 9. Two inputs give the same output.
Fix: restrict the domain (e.g. x ≥ 0) so each output comes from exactly one input.
Restricting the domain for f(x) = x²
f(x) = x² with domain restricted to x ≥ 0. Find f⁻¹(x).
Step by step
- Write y = x² (with domain x ≥ 0, so y ≥ 0).
- Swap x and y.
- Solve for y — take the positive square root (y ≥ 0).
- Write the inverse.
Final answer
f⁻¹(x) = √x, domain x ≥ 0. Without the restriction, f⁻¹ would not be a function.
| Property | f(x) | f⁻¹(x) |
|---|---|---|
| Domain | restricted domain of f | becomes the range of f⁻¹ |
| Range | restricted range of f | becomes the domain of f⁻¹ |
| Rule | domain of f → range of f | range of f → domain of f (reversed) |
[Diagram: math-inverse-function-diagram] - Available in full study mode
IB gives you the restriction — use it: IB will usually tell you the restricted domain in the question.
If the question says "f(x) = x² for x ≥ 0", the restriction is already given.
Your job: find f⁻¹(x) AND state its domain (which equals the range of f).
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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.