Inverse

What is an Inverse?

An inverse undoes a function.

If ff takes you from aa to bb, then f−1f^{-1} takes you from bb back to aa.

f:A→B,f−1:B→Af: A \to B, \quad f^{-1}: B \to A

The Defining Property

f−1(f(a))=a,f(f−1(b))=bf^{-1}(f(a)) = a, \quad f(f^{-1}(b)) = b

Applying ff then f−1f^{-1} (or vice versa) gets you back to where you started.

In terms of composition:

f−1∘f=idA,f∘f−1=idBf^{-1} \circ f = \text{id}_A, \quad f \circ f^{-1} = \text{id}_B

Example

f(x)=x+3,f−1(x)=x−3f(x) = x + 3, \quad f^{-1}(x) = x - 3

Check:

  • f(5)=8f(5) = 8
  • f−1(8)=5f^{-1}(8) = 5

Back to start.

When Does an Inverse Exist?

Only when ff is bijective.

Why not injective?

Two inputs map to the same output — you can’t reverse it.

Why not surjective?

Some output is never hit — f−1f^{-1} has nowhere to send it.

Bijective = invertible.

Example: No Inverse

f(x)=x2 where f:R→Rf(x) = x^2 \text{ where } f: \mathbb{R} \to \mathbb{R}

This is not injective:

  • f(3)=9f(3) = 9
  • f(−3)=9f(-3) = 9

So f−1(9)f^{-1}(9) is undefined — two possible answers.

Fix: Restrict the domain to R≥0\mathbb{R} \geq 0. Then f−1(x)=xf^{-1}(x) = \sqrt{x} works.

Finding an Inverse

To find f−1f^{-1}:

  1. Write y=f(x)y = f(x)
  2. Solve for xx in terms of yy
  3. Swap xx and yy

Example:

f(x)=2x+1f(x) = 2x + 1

  1. y=2x+1y = 2x + 1
  2. x=y−12x = \frac{y - 1}{2}
  3. f−1(x)=x−12f^{-1}(x) = \frac{x - 1}{2}

Key Facts

  • Only bijective functions have inverses
  • (f−1)−1=f(f^{-1})^{-1} = f (inverse of inverse is original)
  • (g∘f)−1=f−1∘g−1(g \circ f)^{-1} = f^{-1} \circ g^{-1} (reverse order)