Finding the domain and range of a composite function

by Future Math person   Last Updated September 11, 2019 09:20 AM

So I have two functions. $f(x) = e^{-x^2+1}$ and $g(x)=\sqrt{x^2-4x+3}$. I am then asked to determine the domain and range of

$a)f∘g,$

$b)g∘f$

I already did part $a)$ and the domain for part $b)$.

For part $a)$, the domain was $(-\infty,1)\cup(3,\infty)$ and the range was $(0,e^2)$.

For part $b$, I figured out that the domain was $(-\infty,-1]\cup[1,\infty)$. I am not sure how to find the range though. Normally, I would take the inverse of g∘f and find the domain of that, and although I can do it, I don't think I did it correctly.

Currently, I did figure out that $g∘f$ is $\sqrt{e^{-2x^2+2}-4e^{-x^2+1}+3}$. How do I find the range of this mess though? I attempted to take the inverse which I believe is:

$y=\pm\sqrt{1-\ln(2\pm\sqrt{1+y^2})}$

Although I know that Wolfram Alpha is not the arbitrator of what correct is, it's generally been right and my answer disagrees with what Wolfram alpha has obtained (As seen here). In addition, the range is something that I am not sure how Wolfram obtained (As seen here). This also looks REALLY messy.

Can anyone guide me as to how this was obtained? That would be much appreciated!



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