About functions and relations

Question

A pure threoretical question here. I have the relation $\pm\sqrt x$. As far as I understand it, that is not a function, as 1 input can map to 2 outputs. If I have a relation, which is not a function, and I limit the inputs, so that every input maps to only one output, can the relation be called a function, altough it's not, for inputs outside of my definition?

Answer 1

Yes, If you restrict the domain and range as you mentioned you can call it a function.

A function is a map from one set of points called the domain, to another set called the range, such that each input maps to exactly one output (note this is not the same thing as requiring that each output corresponds to only one input). The set from which you map does not have to be the entire real domain, in fact it does not even have to be continuous, it could be a discrete set of points; it can be any set you like.

It is good practice to specify the way the function maps the domain to the range. As an example, one way you could make $\sqrt{x}$ a function is if you specify that you are taking positive values of $x$ and that you are taking the positive square root. You can then write the way the map is constructed as follows: $$f:\mathbb R^+\rightarrow\mathbb R^+$$ which says you are building a function that maps from the positive reals to the positive reals. This arrangement is known as the principal square root function.