Question

Find the inverse of the function ? f(x) = x^2*sin(1/x) for x≉0 .

Answer 1

The function does not have an inverse.

Explanation:

In order for a function to have an inverse, it must be injective. As we remember, a function `f:X -> Y` is injective if it maps every element of `X` to a single, unique value of `Y`.

For example, let's have the following function, portrayed through diagrams:

This function is not injective, because `f(3)` and `f(4)` are equivalent, so it is not an one-to-one function.

On the other hand, the one below

is an injective function.

The reason why a function is inversible only if it is injective is due to the definition of functions. A function is a valid function if and only if there exists no `c` such that `f(c)` is multivalued.

If there exists `a` and `b` such that `f(a)=f(b)=alpha`, then the inverse function `f^(-1)` will map `alpha` to both `a` and `b`, which does not make it a function.

For our function, `f(x)= x^2sin(1/x)`, let `gamma = 1/pi` and then `delta=-1/pi`.

`f(color(red)gamma) =(color(red)(1/pi))^2 sin(1/color(red)(1/pi))=1/(pi^2)sinpi=0`

`f(color(red)delta) = (color(red)(-1/pi))^2 sin(1/color(red)(-1/pi))= -1/pi^2sinpi = 0`

Thus, `f` is not injective and, as follows, is not inversible.