108. Inverse Functions

1. Definition

A function $𝑓:𝐴\to 𝐵$ has an inverse function ${𝑓}^{-1}$ if and only if $𝑓$ is bijective (i.e., both one-to-one and onto):

• Injective (One-to-One):

  $𝑓\left({𝑥}_1\right)=𝑓\left({𝑥}_2\right)$ implies ${𝑥}_1={𝑥}_2$

  No two inputs map to the same output

• Surjective (Onto):

  Every element in $𝐵$ is mapped from some element in $𝐴$

2. Finding Inverse Function

To determine ${𝑓}^{-1}$:

• Express $𝑦$ in terms of $𝑥$: $𝑦=𝑓\left(𝑥\right)$
• Solve for $𝑥$ in terms of $𝑦$
• Swap $𝑥$ and $𝑦$, remaining $𝑦$ as ${𝑓}^{-1}\left(𝑥\right)$

3. Graphical Representation

The graph of ${𝑓}^{-1}$ is a reflection of the graph of $𝑓$ access the line $𝑥=𝑦$

2. Derivative of Inverse Functions