Question

What is the relationship between a function and its inverse ?

Answer 1

Hint:We should know that function and the inverse of the function are both different terms but they are also interrelated. We can say that a function in mathematics is an expression, rule or law that defines a relationship between one independent variable to another dependent variable. We should know that the formulas that we use generally are expressions of known functions. As for example, the formula for the area of the circle
$A = \pi {r^2}$, gives the dependent variable $A$ i.e. area as function of the independent variable $r$ (radius).

Complete step by step answer:
We can define a function as a unique relationship between each element of one set with only one element of another set. We commonly symbolize this relationship as:
$y = f(x)$
We know that an inverse function is also known as the anti- function which is defined as the function which can reverse into another function.

That means if the function accepts a certain value or performs the particular operation on the values and generates an output, then the resultant is the inverse of the function. Let us assume that if we are given the function i.e. $y = f(x)$, then the inverse of the function would be $x = {f^{ - 1}}(y)$. So we can say that the inverse function is the function such that if we take the positive sign then the negative sign will be the inverse of it.

There are some interesting facts about inverse and its functions:
-The domain of the function will be the range of the inverse and the range of the function will be the domain of its inverse .
-We should know that the graph of the inverse function will be obtained as the reflection of the graph of the original function across the line $x = y$.

Hence these are the relationship between the function and its inverse.

Note:Let us take an example to understand this. We assume that the function is
$f(x) = 2x + 3$ and we have to find the inverse of this function. Therefore let us assume that $y = f(x)$. So we can write;
$y = 2x + 3$
So here we can calculate the value of $x$ by transferring the constant term to the left hand side:
$y - 3 = 2x$
By isolating the term $x$ we have:
$x = \dfrac{{y - 3}}{2}$
Here we get the value of $x$ in terms of $y$.
As we know that
$y = f(x)$, then we can write that
$x = {f^{ - 1}}(y)$
So we can write this as
${f^{ - 1}}(y) = \dfrac{{y - 3}}{2}$
Or we can say that
${f^{ - 1}}(x) = \dfrac{{y - 3}}{2}$
So this is the required inverse of the given function.