Invertible Functions: Definition, Test, and Examples

A function is invertible when it can be perfectly undone: there is a reverse function f inverse that turns every output back into its original input. This happens exactly when f is bijective — both one-one and onto.

It matters because "undo" is everywhere: decoding a message, converting Fahrenheit back to Celsius, or reversing any calculation all need the inverse function. Without bijectivity, undoing is ambiguous or impossible.

Key Concepts

• Invertible function: a function f: A to B for which there exists g: B to A with g(f(x)) = x for all x in A and f(g(y)) = y for all y in B.
• Inverse f inverse: that reverse function g, unique when it exists.
• Bijectivity test: f is invertible if and only if f is one-one and onto.
• Graphical view: the graph of f inverse is the mirror image of the graph of f across the line y = x.

Formulas

• f inverse(f(x)) = x and f(f inverse(y)) = y: applying a function and its inverse in either order returns the starting value.
• (f inverse) inverse = f: the inverse of the inverse is the original function.
• (g circle f) inverse = f inverse circle g inverse: invert a chain by reversing the order.
• Finding the inverse: write y = f(x), solve for x in terms of y, then swap the letters to get y = f inverse(x).

Worked Example

Find the inverse of f: R to R with f(x) = 2x + 3, and verify it.

1. Set y = 2x + 3 and solve for x: 2x = y minus 3, so x = (y minus 3)/2.
2. Swap letters: f inverse(x) = (x minus 3)/2.
3. Verify f inverse(f(4)): f(4) = 11, and (11 minus 3)/2 = 4. Back to the start.
4. Verify f(f inverse(7)): f inverse(7) = 2, and f(2) = 7. Both directions undo cleanly, so the inverse is correct.

Common Misconceptions

• "f inverse means 1/f." Never: (x minus 3)/2 is the inverse of 2x + 3, while 1/(2x + 3) is the reciprocal — a completely different function.
• "Every function has an inverse." Only bijective ones do. f(x) = x squared on R to R fails both one-one and onto, so no inverse exists there.
• "Restricting the domain is cheating." It is standard practice: x squared becomes invertible once restricted to x greater than or equal to 0, giving the square root.

Real-World Applications

• Decryption: the decryption key is the inverse function of the encryption function.
• Temperature scales: F = (9/5)C + 32 inverts to C = (5/9)(F minus 32).
• Undo buttons: every Ctrl+Z applies the inverse of the last editing operation.

A function is invertible exactly when it is bijective, and its inverse undoes it in both directions: f inverse(f(x)) = x. Solve y = f(x) for x to find the inverse, and never confuse it with the reciprocal 1/f.