If a function is a machine that turns inputs into outputs, its inverse is the machine run backwards — it takes the original outputs and hands back the original inputs. If , then . Inverses show up constantly: converting Celsius to Fahrenheit and back, encrypting and decrypting a message, or finding "how many hours to reach a distance" from "distance as a function of hours." Whenever a question gives you the output and asks for the input over and over, the inverse function answers all of those questions at once.
Finding an inverse algebraically
- Replace with .
- Swap and everywhere in the equation.
- Solve the new equation for .
- Replace with (read " inverse of " — this is notation, not an exponent ).
Why does swapping work? The graph of is the set of points . The inverse should contain the same pairs with the roles reversed, — and swapping and in the equation does exactly that. There is also a second way to see it: an inverse undoes the steps in reverse order, like taking off your shoes before your socks. means "multiply by 2, then subtract 3." Undoing it means "add 3, then divide by 2," which is . Both methods always agree; the reverse-steps view is a quick mental check.
Reflection in y = x
Swapping and is exactly the algebraic version of reflecting the graph across the line . Every point on becomes the point on — so the graphs of and are always mirror images of each other across . This gives a fast way to sketch an inverse without any algebra: draw the dashed line , take three or four easy points on , swap each point's coordinates, and join the new points. For , the points , and become , and on . Notice the slopes: has slope and has slope — a linear function's inverse always has the reciprocal slope, because rise and run trade places.
Three consequences follow. First, the domain and range swap: the domain of is the range of , and the range of is the domain of . Second, any point where meets the line (like on ) lies on both graphs, since swapping its coordinates changes nothing. Third, applying a function and then its inverse returns you to the start: and . That last fact is the standard way to verify an inverse — substitute one into the other and check that everything simplifies to .
When is the inverse a function?
Swapping and always produces a valid relation, but not always a valid function. Since the inverse relation is the reflection of , the vertical line test on the inverse is equivalent to a horizontal line test on the original: if any horizontal line crosses 's graph more than once, then fails the vertical line test and is not a function.
This is exactly what happens with — it's a parabola, so a horizontal line like crosses it twice (at and ). Run backwards, the output would have to return two inputs, which a function can't do. To fix this, we restrict the domain of the original function (e.g. to ) so that it passes the horizontal line test, which guarantees the inverse is a genuine function. For a parabola the natural cut is at the vertex: keep either the right half or the left half, and choose the matching sign of the square root.
| Function | Restricted domain | Inverse | Inverse domain |
|---|---|---|---|
| none needed |
Inverses in context
In applications the variables have meanings, so it often makes more sense to solve for the original input than to literally swap letters. Temperature in Fahrenheit is . Solving for gives the inverse, , which converts the other way: °F becomes °C. A taxi fare (in dollars, for kilometres) has inverse , which tells you how far a given budget will take you. Keep the variable names — swapping and would mix up the units. Context also explains when an inverse fails. A ball's height rises and then falls, so a height of m happens twice — once going up and once coming down. Asking "at what time is the height 10 m?" has two answers, so the inverse is not a function unless you restrict to the rising part (before the vertex) or the falling part (after it). That is the horizontal line test in real-world clothing.