Functions definition + question Am I correct in saying that for Functions, the below is the correct definition:
For each value of x in the domain there is only one value of y in the range.
Hence, the picture below means that it is not a many-to-one function as the values of x do not map onto a single value of y in the range (they map onto two values of y in the range) and therefore it is not a function but an operation (square root)

 A: Yes the picture depicts a one to many mapping which is NOT a function. 
A: A function $f:X \to Y$ is a subset $A\subset X × Y$ subject to the condition that every element of $X$ is the first component of one and only one ordered pair in the subset $A$. Hence, the picture depicts a one-to-many mapping - this is not a function.
A: A function is a relation between a set of inputs and a set of permissible outputs (determined by the relation), with the property that each input is related to exactly one output. So you need to make sure to change "only" to "exactly" or to " one and only one".
I emphasize one and only one because each input value must be related to one output value, and only one output value.
The picture cannot depict a function from the set on the left to the set on the right, for the reason you give.  However, if the direction of all the arrows were reversed, then we would have a function from the set on the right to the set on the left, though it would not be a one-to-one function.
