To reverse a function, undo its operations in the opposite order. A mapping can be reversed cleanly only if each output comes from exactly one input, which is what “one-to-one” means.
This lesson follows tracing an input through a composite function and belongs to functions and mappings. It also appears, in disguise, whenever you work backwards from a final value to an original one.
What does “reverse” mean?
Take the mapping x → 3x + 1 on the inputs 1, 2 and 3.
| Input x | Output f(x) |
|---|---|
| 1 | 4 |
| 2 | 7 |
| 3 | 10 |
Reversed, 4 goes back to 1, 7 to 2 and 10 to 3. The reverse mapping reads the table from right to left. Because no two inputs share an output, nothing is lost by reversing.
How to find an inverse, step by step
- Write y = the rule. For example y = 3x − 5.
- Make x the subject. Undo the last operation first: add 5, then divide by 3.
- Rename: replace y with x in the result and write it as f⁻¹(x).
- Check with a number: apply f, then f⁻¹, and confirm you return to the start.
Worked example
Find the inverse of f(x) = 3x − 5.
Step 1: y = 3x − 5.
Step 2: y + 5 = 3x, so x = (y + 5)/3.
Step 3: f⁻¹(x) = (x + 5)/3.
Check: f(4) = 12 − 5 = 7. Then f⁻¹(7) = (7 + 5)/3 = 12/3 = 4. We are back at 4, so it works.
The same idea appears in percentages.
A price after a 20% increase is 1.2 times the original, so x → 1.2x. If the new price is RM96, the original is found by the inverse, dividing by 1.2: 96 ÷ 1.2 = RM80.
Check: 80 × 1.2 = 96. The percentage-base explorer shows why you divide rather than subtract 20%.
The mistake to watch for
Two slips are common. One is to take the reciprocal, and the other is to undo the operations in the wrong order.
Mistaken working: f(x) = 3x − 5, so f⁻¹(x) = x/3 + 5
The student undid the operations in the same order as they were applied, dividing by 3 before adding 5.
Test it: f⁻¹(7) would give 7/3 + 5, which is not 4.
The correct approach is to undo the last operation first. f did ”× 3, then − 5”, so the inverse does ”+ 5, then ÷ 3”. A single test number exposes the error in seconds.
Check yourself
Work on paper first, then open each answer.
1. Find the inverse of f(x) = 2x + 7, then evaluate f⁻¹(11).
Show answer
y = 2x + 7, so x = (y − 7)/2. Therefore f⁻¹(x) = (x − 7)/2. f⁻¹(11) = 4/2 = 2. Check: f(2) = 11.
2. Find the inverse of f(x) = x/4 − 3.
Show answer
y = x/4 − 3, so y + 3 = x/4, and x = 4(y + 3). Therefore f⁻¹(x) = 4(x + 3), which is 4x + 12. Check: f(8) = 2 − 3 = −1, and f⁻¹(−1) = 4(2) = 8.
3. Find f⁻¹(−3) when f(x) = 5 − 2x.
Show answer
y = 5 − 2x, so 2x = 5 − y and x = (5 − y)/2. So f⁻¹(x) = (5 − x)/2, and f⁻¹(−3) = 8/2 = 4. Check: f(4) = 5 − 8 = −3.
Where this leads next
Not every rule can be reversed over all numbers, so next look at restricting inputs to keep an expression defined. The function composition and inverse explorer and the non-calculator working trainer are useful for testing your inverses with numbers.
If undoing operations in the right order still feels uncertain, our teachers can work through it with you in online one-to-one Mathematics tuition.