How are $y=\sqrt{x}$ and $y^2=x$ different functions?

Question

Is $$ y = \sqrt{x} $$

Any Different From $$ y^2 = x $$

If we square the function $y=\sqrt{x}$ , then don't we obtain $y^2=x$

Which means these are same functions but they are not?

How does this happen?

Answer 1

The function $$ y = \sqrt{x} $$

Is defined only for positive values of y.

But The Function $$ y^2 = x $$

Is defined for all Real values of y.

Moreover, you could see their graphs .

Answer 2

No, they are not.

Squaring both sides does not always work. Consider:

-2 and 2, they are not equal bit their squares are.

$$y^2=x$$ is the same as $$y = \pm \sqrt{x}$$

Answer 3

The relation $R=\{(x,y)|y^2=x\}$ is not function because for example $(4,2)\in R$ and $(4,-2)\in R$, but $2\neq-2$.

By the way, the relation $F=\{(x,y)|y=\sqrt{x},x\geq0,y\geq0\}$ is a function.