When does an inverse exist?
Learn to spot repeated outputs, choose a domain, and keep the correct inverse branch.
By the end, you can
- Explain why a function is one-to-one
- Restrict a domain so an inverse exists
- Find an inverse with its domain and range
about 20 minutes for the lesson · practise at your own pace.
The central idea
A function can be reversed when each output identifies exactly one allowed input. This is the one-to-one condition. For on all real numbers, . Knowing the output is 4 does not tell us which input was used. A horizontal line crossing the graph twice reveals the same problem.
Worked example
Keep only . The function is now one-to-one. Write . Since the original input is nonnegative, . Swap the letters to obtain for . The domain of the inverse is the range of the original function: . If we instead kept , the inverse would be on the same domain.
Swapping x and y does not fix a function that is many-to-one. Choose a domain first, then keep its sign and endpoint conditions.
Check your understanding
For , restricted to , find and its domain. Why must you choose the positive square root?
, with domain . The original input satisfies , so rearranging must use the positive square root. The inverse has range .
Practice questions
12 examples and practice questions, in 2 sections.
More practice
Begin with a familiar inverse, then work through more demanding domain restrictions.
- When can two functions compose?Part (ii) inverts a square root whose domain is already restricted.Open question and interactive solution →
- Restrict the domain to choose an inverseChoose the boundary that makes a bell-shaped function invertible.Open question and interactive solution →
- Absolute values and the correct inverse branchApply the same idea when an absolute value creates several branches.Open question and interactive solution →
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