Inverse function or reciprocal? The −1 notation has two different jobs
f⁻¹ reverses a function's input-output pairing; 1/f takes reciprocal outputs. They are generally different operations.
A practical comparison that turns a vague choice into a repeatable test.
Updated July 13, 2026The superscript −1 on a function name means inverse function, not a negative exponent applied to the output.
An inverse reverses the mapping
If f sends input a to output b, then f⁻¹ sends b back to a. The domain and range exchange roles.
A function needs to be one-to-one on the chosen domain to have an inverse function. The horizontal-line test checks this on a graph.
A reciprocal changes output values
The reciprocal function 1/f(x) keeps the same input but replaces each nonzero output with its multiplicative reciprocal.
Its domain excludes zeros of f. It does not undo the original function.
Solve for the inverse carefully
Write y = f(x), exchange x and y, and solve for y. Then verify both compositions where domains allow.
Domain restrictions may need to be added, especially when reversing squares, roots, or rational functions.
For f(x) = 3x − 6, find f⁻¹(x) and 1/f(x).
Start with the original rule.
Exchange x and y, then solve for y.
State the inverse function.
Take the reciprocal separately and state its restriction.
Common mistakes
- Replacing f⁻¹(x) with 1/f(x).
- Swapping x and y without solving for y.
- Ignoring one-to-one or domain requirements.
Three takeaways
- Inverse functions undo mappings.
- Reciprocals invert output values.
- Verify inverses by composition.