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I definitely think of this question as a very fundamental one. But I'd like to figure out what the image of exp is in general.

Given a polynomial function $f(x)$, how can we determine the image( plot, precisely) of

$\exp({f(x)})$??

Thanks to two earlier comments. I append more details what i want: According to your explanations, we can find out just the range. What about the shape of exp(f(x))??

mallea
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  • If you want to see it, graph it once. They all look the same. (I always thought they look like slugs.) If you want to "get" them ... well there's a lot to say, mainly because the are their own derivatives. They are convex smooth and scalable. – fleablood Feb 19 '17 at 08:41
  • Oh, "image" in the mathematical sense! The range that is strictly mapped to. It's (0,infinity). If b>0, b!= 1 then b^x is always positive, asymptotic to zero in one direction and unbounded in the other. – fleablood Feb 19 '17 at 08:44

2 Answers2

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I presume you are talking about real numbers. Find the image of $f$. If that is $[a,\infty)$, then the image of $\exp(f)$ is $[\exp(a),\infty)$. If it is $(-\infty, b]$, then $(0, \exp(b)]$. If it is $(-\infty, \infty)$, then $(0,\infty)$.

Robert Israel
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First you need to determine the range of $f(x)$, which is of one of the forms $(-\infty,a]$, $[b,+\infty)$ or $(-\infty,+\infty)$. Then the range of $\exp(f(x))$ will be $(0,e^a]$, $[e^b,+\infty)$ or $(0,+\infty)$, accordingly.

user49640
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