Definition:

An inverse function reverses the operation of the original function. An inverse function in algebra is a function that "undoes" the action of another function. For a function \(f(x)\) its inverse, denoted as \(f^{-1}(x)\), “undoes” what \(f(x)\) does. If you have a function ( f(x) ), its inverse, satisfies the following condition: \[ f(f^{-1}(x)) = x \quad \text{and} \quad f^{-1}(f(x)) = x \] Specifically, if \(f(a) = b \) then \(f^{-1}(b) = a \)

The inverse function of a function \( f(x) \) is denoted as \( f^{-1}(x) \) and satisfies the property: \[ f(f^{-1}(x)) = f^{-1}(f(x)) = x \]

In simpler terms, applying a function and then its inverse (or vice versa) will bring you back to the original input. For the inverse function to exist, the original function ( f(x) ) must be bijective—that is, both one-to-one (injective) and onto (surjective).
To have an inverse, a function must satisfy the condition that each input has a unique output and vice versa. Therefore, only one-to-one functions have inverses because they meet this requirement.
Why One-to-One inverse function?
If a function is not one-to-one, it doesn’t pass the horizontal line test, meaning some output values map to multiple input values. This ambiguity makes it impossible to define a single unique inverse
For example:
If \(f(x) = x^2\), both - 2 and 2 produces the same output, that is , \( f(-2) = f(2) = 4\)
Therefore, \(f^{-1}(4)\) would be both -2 and 2, which contradicts the definition of a function (each input must map to a unique output).

Mathematical definition:

If 𝑓 is a one-to-one function, then 𝑓 has an inverse function \(f^{-1}\) such that:

1. \[f(f^{-1}(x)) = x \]

   for all \(x\) in the domain of \(f^{-1}\)

2. \[f^{-1}(f(x)) = x \]

   for all \(x\) in the domain of \(f\)

Inverse Function: Graphing Technique

To graph an inverse function \(f^{-1}\), we can use the following techniques:

1. Switch Coordinates of Points
   To plot the inverse function is by switching the 𝑥- and 𝑦-coordinates of each point on the graph of 𝑓(𝑥).
   Example:
   If a point (a, b) lies on \(f(x)\), then the point (b, a) will lie on \(f^{-1}(x)\) which can be seen in the following first graph.
2. Reflect Across the Line \(y = x\)
   The graph of an inverse function is a reflection of the original function across the line \(y = x\) as in the following second graph . Example:
   The graph of the function \(f(x) = 2x + 3 \) is a line. The graph of the inverse function \(f^{-1}(x)\) is also a line, and both lines are reflections of each other across \(y = x\).

$$ f(x) = mx + b $$