2.5 Inverse of a function

Inverse relation is like looking a relation in opposite direction. Equivalently, it is also like an image in a mirror. For example, consider the relation “husband of”. The inverse to this relation is “wife of”. This is an explicit relation very easily conceivable. In other situations involving function, inverse relations may not be so explicit. We shall, therefore, develop mathematical technique to obtain inverse function (relation) for a given function (relation).

In order to facilitate easy recapitulation of concepts and terms for the study of inverse relation, we can refer meaning attached to following terms :

• $f:$ denotes function relation from domain “A” to co-domain “B” of “f” (implicit inference of sets).
• $f:A\to B:$ denotes function relation from domain “A” to co-domain “B” of “f” (explicit reference of sets).
• $f\left(x\right):$ denotes image, an instance of function “f”, image under “f”, rule of “f”.
• $f^{-1}:$ denotes inverse function relation from domain “A” to co-domain “B” of “ $f^{-1}$ ” (implicit reference of sets)
• $f^{-1}:A\to B:$ denotes inverse function relation from domain “A” to co-domain “B” of “ $f^{-1}$ ” (explicit reference of sets)
• $f^{-1}\left(x\right):$ denotes pre-image, an instance of function “ $f^{-1}$ ”, image under “ $f^{-1}$ ”, rule of “ $f^{-1}$ ”

Inverse of an element

We use the concept of pre-image and image to connect the elements of a function in the direction from domain “A” to co-domain “B”. The related elements are connected by a rule “f(x)” such that :

$$\text{Image}=f\left(x\right)=f\text{(pre-image)}$$

Clearly, “x” is the pre-image and “f(x)” is image. Now, we want to derive a similar rule, " $f^{-1}\left(x\right)$ ", which evaluates to pre-image like :

$$\text{Pre-image(s)}=f^{-1}\left(x\right)=f^{-1}\text{(image)}$$

Clearly, “x” is now the image and “ $f^{-1}\left(x\right)$ ” is pre-image(s). The important point to understand here is that the image in the co-domain set can be related to none, one or more elements in domain set. Therefore, this rule may evaluate accordingly to value(s) – none, one or more - for the pre-images.

Method to construct an inverse rule

We construct an inverse rule in step-wise manner as enumerated here with an example :

Step 1: Write down the rule of the given function “f”.

Let the given rule be f(x) given by :

$$f\left(x\right)=x^2$$

Let us put y = f(x). Then,

$$\Rightarrow y=f\left(x\right)=x^2$$

This relation gives us one value of image. For example, if x = 3, then

$$\Rightarrow y=3^2=9$$

Step 2: Solve for “x”

$$\Rightarrow x=\pm \sqrt{y}$$

Step 3: Replace “x” which represents pre-image by the symbol “ $f^{-1}\left(x\right)$ ” and replace “y” which represents image by “x”. For the given function, $f\left(x\right)=x^2$ , the new inverse rule is :

$$\Rightarrow f^{-1}\left(x\right)=\pm \sqrt{x}$$

This is how we construct the inverse rule. Note emphatically that “x” now represents “image” and “ $f^{-1}\left(x\right)$ ” represents “pre-image”. For example, if image is “9”, then we can find its pre-image (s), using this new rule as :

$$\Rightarrow f^{-1}\left(9\right)=\pm \sqrt{9}=\pm 3$$

Thus, the required pre-images is a set of two pre-images :

$$\Rightarrow f^{-1}\left(9\right)=\{-\mathrm{3,}3\}$$

Inverse of a function

Once the inverse rule is constructed, it is easy to define inverse function. However, we should be careful in one important aspect. An inverse function, “ $f^{-1}$ ” is a function ultimately. This puts the requirement that every element of the domain of the new function “ $f^{-1}$ ” should be related to exactly one element to its co-domain set.