Algebra 2 · Unit 1 · Topic 1.4
Inverse Functions and Domain Restrictions
An inverse function reverses a one-to-one function's input-output pairs.
Definition and explanation
An inverse function reverses a one-to-one function's input-output pairs. To find a formula, write y = f(x), exchange x and y, and solve for y; verify the compositions on the relevant domains.
Worked example
For f(x) = 5x − 9, exchange variables in y = 5x − 9 to get x = 5y − 9. Hence f⁻¹(x) = (x + 9)/5. The check f(f⁻¹(x)) = 5(x + 9)/5 − 9 = x holds for all real x.
Common mistake
The inverse notation f⁻¹ is not the reciprocal 1/f. The function x² needs a restricted domain to have an inverse function.
Key takeaway
An inverse function reverses a one-to-one function's input-output pairs.
Explain each step of the worked example, then check the units, signs, and conditions that apply.
Further reading
Topic coverage was checked against the openly available OpenStax Intermediate Algebra 2e, Algebra and Trigonometry 2e, and Introductory Statistics 2e textbooks. The explanations and examples on this site are independently written; textbook prose, exercises, and artwork are not reproduced.