The basics

$-4$ is not the correct answer to the problem above.

Function: A function is a relation that associates a member of a set $X$ (called the domain) to a unique member of another set $Y$ (called the co-domain). It can be denoted as $y=f(x)$ . More than one member of the domain can be related to a single member of co-domain but the converse is not true.

Let $y = x^2$ be a function. If you have studied coordinate geometry you will notice that this is the equation of a simple parabola. Here $x$ can take both positive and negative values (of the same magnitude) for each positive $y$ . Hence for a given $y$ , we have two solutions for $x$ .

Now let $y = \sqrt{x}$ be an expression. This is not a proper function unless we define what value $\sqrt{x}$ will give. Further, this value should be 'unique' to categorize it into a function. To avoid this ambiguity, the principal square root is taken to be the positive root . Hence for a given $x$ there is a unique solution to $y$ .

For any real $a$ , we thus define $\sqrt{a^2} = |a|$ .