Apologies, this is a simple question but I seem to have some sort of brain freeze. I'm looking for the range of this piecewise function:
$$f(x)=\begin{cases}x+9&\text{ if } x<-3\\
-2x&\text{ if }-3\leq x\leq 3\\
-6&\text{ if }x>3\end{cases} $$
The domain is $\mathbb{R}$ and is the range $[-6,6]$ or $(-\infty, -6]$?
Thanks
Best Answer
If $x<-3$ we get $f(x)<6$ and $f(-3)= -2(-3) =6$ so $(-\infty,6]\subseteq Range(f)$.
We get nothing new from other parts of function. So $Range(f)= (-\infty,6]$ since it is continuous on $(-\infty ,-3]$.