Range of a piecewise function

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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

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1 Answer

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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 $(-\infty ,-3]$.

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