Inverse Function

Understanding Inverse Functions

You often translate words or sentences from English to Indonesian, for example, when watching movies or reading news. This translation process is similar to how a function works: there is an input (English word) and an output (Indonesian word).

Consider the following illustration:

Here, the "Translation Machine" acts like a function that transforms "Mathematics" (input) into "Matematika" (output).

Now, what if we want to do the reverse? Translate "Matematika" back into "Mathematics"?

This reverse process is the basis of the inverse function concept.

Definition of Inverse Function

An inverse function is a function that "reverses" the operation of an initial function. If function $f$ maps element $x$ from domain $A$ to element $y$ in codomain $B$, then its inverse function, denoted as $f^{-1}$ (read "f inverse"), maps element $y$ from $B$ back to element $x$ in $A$.

Mathematically:

In other words, if $f$ changes $x$ to $y$, then $f^{-1}$ changes $y$ back to $x$. The inverse function "undoes" the effect of the original function.

Important: The notation $f^{-1}(x)$ does not mean $\frac{1}{f(x)}$ (multiplicative inverse or reciprocal). It is special notation for the inverse function.

Condition for a Function to Have an Inverse

Not all functions have an inverse function. For a function $f$ to have an inverse function $f^{-1}$, the function $f$ must be bijective. A bijective function is both injective (one-to-one) and surjective (onto).

• Injective (One-to-one): Every distinct element in the domain maps to a distinct element in the codomain. No two different inputs produce the same output.
• Surjective (Onto): Every element in the codomain is the result of mapping from at least one element in the domain. All possible outputs occur.

If function $f$ is not bijective, its inverse relation might exist, but that relation will not be a function.

Determining the Formula for an Inverse Function

To find the formula for the inverse function $f^{-1}(x)$ from a function $f(x)$, you can follow these steps:

1. Replace $f(x)$ with $y$.
2. Swap the positions of the variables $x$ and $y$ in the equation.
3. Solve the equation for $y$ in terms of $x$.
4. Replace $y$ with $f^{-1}(x)$ to get the inverse function formula.

Example:

Find the inverse function of $f(x) = 2x + 3$.

1. Replace $f(x)$ with $y$:

   $$y = 2x + 3$$

2. Swap $x$ and $y$:

   $$x = 2y + 3$$

3. Solve for $y$:

   $$x - 3 = 2y$$

   $$\frac{x - 3}{2} = y$$

4. Replace $y$ with $f^{-1}(x)$:

   $$f^{-1}(x) = \frac{x - 3}{2}$$

So, the inverse function of $f(x) = 2x + 3$ is $f^{-1}(x) = \frac{x - 3}{2}$.

Graph of a Function and Its Inverse

The graph of the inverse function $f^{-1}(x)$ is a reflection of the graph of the original function $f(x)$ across the line $y = x$.

For example, let's look at the graph of $f(x) = 2x + 3$, its inverse $f^{-1}(x) = \frac{x - 3}{2}$, and the identity line $y=x$.