How can we figure out the image of exponential function?

Question

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

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

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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))??

Answer 1

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 $(0,\infty)$.

Answer 2

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.