single valued function and multi valued function square root function is a multi valued function. for example 16 have two square root 4 and -4. but it is violation of the definition of function that each element of domain has one and only one image. then how it is a function. please clear my dout.
 A: 
for example 16 have two square root 4 and -4.

Yes, that is correct.

square root function is a multi valued function.

Not quite.  Square roots are different from the square root function.  The definition of $\sqrt{x}$ is not all numbers $y$ satisfying $y^2=x$, but the unique nonnegative number $y$ such that $y^2=x$.  
Thus $\sqrt{16} = 4$, not $4$ and $-4$.  
With this restriction, $f(x) = \sqrt{x}$ is a function with domain $[0,\infty)$.
A: You clearly understand that a positive real number has two real square roots. I think your question is really about language. The phrase "multi-valued function" does indeed suggest that "multi-valued" is an adjective modifying the noun "function", thus describing a particular kind of function. Which, as you point out, would violate the very definition of "function". That's a reasonable grammatical way to look at it. But it's not right. The phrase has to be read all in one piece. A "multi-valued function" is not a function, it's a different kind of object, one carefully studied when you get to complex (as opposed to) real analysis.
There are other places where mathematicians twist grammar this way. One is "manifold with boundary". For an everyday usage think "plastic glass".
By the way, none of this applies to the real valued function written with the symbol $\sqrt{}$. That's not multi-valued. By convention it always tells you the non-negative root.
A: When we're doing real number arithmetic, the square root function means the positive square root.  $16$ does have two square roots, but only one of them is the output of the function.
A: $f:[0,\infty)\to[0,\infty)$ defined by $f(x)=\sqrt{x}$ assigns to each $x$ a single value, the positive square root. This is a function.
Similarly, $-f$ is a function. 
Functions cannot output more than one value for a given input.
