Inverse Function

Definition

An inverse function reverses another function.

If:

f(a) = b

then the inverse sends:

b -> a

Inverse function = function that undoes another function.

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Simple Example

Let:

f(x) = x + 1

Then the inverse is:

f^-1(x) = x - 1

Example:

f(3) = 4
f^-1(4) = 3

The inverse takes the output back to the original input.

An inverse reverses direction.

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Inverse Requires No Ambiguity

Not every function has an inverse function.

Example:

f(x) = x * x
Domain: {-2, -1, 0, 1, 2}

Then:

-2 -> 4
2 -> 4

If the output is 4, the original input could be -2 or 2.

The reversal is ambiguous.

If different inputs collapse into one output, the inverse is not a function.

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Connection To Bijective Functions

A function has a clean inverse function when it is bijective.

Bijective means:

no two inputs share an output
and every codomain value is reached

This gives every output exactly one input to return to.

Bijectivity makes inverse functions well-defined.

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Undoing With Composition

If f^-1 is the inverse of f, then applying both returns the original value.

Example:

f(x) = x + 1
f^-1(x) = x - 1

Then:

f^-1(f(3)) = 3
f(f^-1(4)) = 4

A function and its inverse cancel each other.

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Final Summary

• An inverse function reverses another function.
• If f(a) = b, the inverse sends b back to a.
• Not every function has an inverse function.
• If multiple inputs produce the same output, reversal is ambiguous.
• Bijective functions have clean inverse functions.
• A function and its inverse undo each other.