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.
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 $[0,\infty)$.
$f:[0,\infty)\to[0,\infty)$ 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.