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Say that I have a function $f(x)$ with range $(-\infty,10]$ and a function $g(x)$ where $g(x)=2f(-x)-3$, what is the range of $g(x)$?

So do I do the vertical shift first or do I do the vertically stretching first? I can't just stretch it first and then shift down 3 and have the new range be $(-\infty,17]$? Does the stretching factor matter? Or should it just be $(-\infty,7]$?

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Since $\varphi:\mathbb{R}\to\mathbb{R},x\mapsto -x$ is a bijection, we see that $f$ and $f\circ\varphi:x\mapsto f(-x)$ have the same range, i.e. $(-\infty,10]$

Remark : $f$ and $f\circ\varphi$ don't have, however, the same domain; except if we suppose that the domain of $f$ is symmetrical with respect to $0$.

Next, let $u:(-\infty,10]\to\mathbb{R},x\mapsto 2x-3$. Since $u$ is continuous and increasing, its range is $(-\infty,2\times 10-3]=(-\infty,17]$.

Finally, $g=u\circ f\circ\varphi$ has range $(-\infty,17]$.

Here is an illustration :

enter image description here

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  • $\begingroup$ that is such a cool graph, thank you so much! $\endgroup$ – Akaichan Feb 16 '17 at 11:03

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