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.
Best Answer
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.