Question

How would you find the inverse of $y = {x^2}$ and is it a function?

Answer 1

Here we must know that whenever we are given the function as $y$ in the terms of $x$ and we need to find the inverse, we actually need to find $x$ in terms of $y$ and then we will get inverse of that function be replacing at last in inverse $y{\text{ by }}x$ and we must know that for every domain if there is a single element in the codomain then it is a function otherwise not.

Complete step by step solution:
Here we are given to find the inverse of the function $y = {x^2}$
So we are given $y = f\left( x \right)$ as it is the function of $x$ and now we know that whenever we need to calculate the inverse we just need to find $x$ in terms of $y$
So we can say that $y = {x^2}$
Therefore $x = \pm \sqrt y$
As $x$ is the inverse of $y$ we can put $x = {f^{ - 1}}\left( y \right)$
${f^{ - 1}}\left( y \right) = \pm \sqrt y$
Replacing $y{\text{ by }}x$
${f^{ - 1}}\left( x \right) = \pm \sqrt x$

Hence we get the inverse as $\pm \sqrt x$

Now we know that for every domain if there is a single element in the codomain then it is a function otherwise not.
Here we can see that for every value of $x$ we will get two values in the codomain one positive and one negative. Hence it does not have one value. So it is not a function.

Note:
Here if we would have only $\sqrt x$ instead of $\pm \sqrt x$ then it would be a function as then there would be only a single value in the codomain for every value in the domain.