Run the machine backwards: If a function turns an input into an output, its inverse f⁻¹ turns that output back into the original input.
So f⁻¹(f(x)) = x: do something, then undo it, and you are back where you started. Doubling is undone by halving; adding 3 is undone by subtracting 3.
Note f⁻¹ is the inverse, not '1 divided by f'. It is a brand-new function that reverses the steps of f in the opposite order.
IB-style question — read an inverse value
A water tank's volume after x minutes of filling is f(x) = 3x + 5 litres.
Find f⁻¹(20) and explain what it means.
Step by step
- f⁻¹(20) asks: which input gives an output of 20? Set f(x) = 20.
- Undo the steps: subtract 5, then divide by 3.
Final answer
f⁻¹(20) = 5. In context: it takes 5 minutes for the tank to reach 20 litres.
IB-style question — undo, in the right order
g(x) = 2x − 7 takes an input, doubles it, then subtracts 7.
Describe the steps of g⁻¹.
Step by step
- Reverse the operations in the OPPOSITE order: undo 'subtract 7' first.
- Then undo 'double'.
Final answer
g⁻¹ adds 7, then halves: g⁻¹(x) = (x + 7)/2. (Last operation done by g is undone first.)
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Swap x and y, then make y the subject: The reliable recipe:
1. Write y = f(x).
2. Swap every x and y.
3. Solve the new equation for y — that is f⁻¹(x).
Why the swap works: inputs and outputs trade places when you reverse a function, and graphically that is a reflection in the line y = x. The domain and range of f also swap: the range of f becomes the domain of f⁻¹.
IB-style question — find the inverse
A taxi charges f(x) = 4x + 3 dollars for a journey of x km.
Find f⁻¹(x).
Step by step
- Write y = f(x).
- Swap x and y.
- Solve for y: subtract 3, divide by 4.
Final answer
f⁻¹(x) = (x − 3)/4 — it converts a fare back into the distance travelled.
IB-style question — inverse of a rational function
Find the inverse of f(x) = 2/(x − 1), where x ≠ 1.
Step by step
- Write y, then swap x and y.
- Multiply both sides by (y − 1).
- Make y the subject: y − 1 = 2/x, so add 1.
Final answer
f⁻¹(x) = 2/x + 1, with x ≠ 0 (since the range of f never reaches 0).