Piecewise functions "overlap"

Question

The question asks "Decide whether the following is a function or not; justify your answer." Then they give the following piecewise "function": $$ f(x) = \begin{cases} x+1, & \text{if } x < 0\\ \sqrt{x+3}, & \text{if } x > -3 \end{cases} $$

Now so far I have the following sketch:

What I want to know is that in the case of $\sqrt{x+3}$ does it need to be drawn from $x>-3$ (indicated with red) or from $\sqrt{3}$.

I know that, if the part I have indicated in red should not be included, it will be a function. But if it stays it is not a function.

Answer 1

Your relation is not a function, since given any point $x \in (-3,0)$, the quantity $f(x)$ is not uniquely defined.

You are right, if you exclude this interval from any one of the branches, it will become a function.

The way the problem is formulated, to draw its graph you need to indeed draw both branches the way you showed.

Answer 2

Why shouldn't the part in red be included? That graphs exactly what happens on the second leg of the proposed definition when $x > -3$! So yes, the combined clauses give two values to the "function" between $-3$ and $0$. So, on modern definitions, it isn't a function.

It is always worth noting though, as a footnote, that the insistence that functions (properly so-called) be single-valued is a relative late-comer. G.H. Hardy, for example, in his once-canonical A Course in Pure Mathematics explicitly and emphatically says that the characteristic of taking just one value for a given argument "is by no means involved in the general idea of a function".