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There's the standard transformations of f(x) that you get taught in school, such as nf(x) or f(nx), but no teacher has ever mentioned f(x) to a power.

So how do you do $(f(x))^n$?

For example, how would you sketch the graph of $(cosx)^{sinx}$?

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    $\begingroup$ Good question to ask. IMO, there is no easy way to do that. The best that I know of, is to do a point wise consideration. I even advocate for point wise consideration during 'standard' transformations, esp for students who do not really understand what is happening, or why. $\endgroup$ – Calvin Lin Feb 23 '14 at 12:58
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    $\begingroup$ This has nothing to do with graph theory. $\endgroup$ – Henning Makholm Feb 23 '14 at 13:05
  • $\begingroup$ OP, for future reference, graph theory is this. Also, your example doesn't seem to match your question. That's not $f(x)^n$, it's $f(x)^{g(x)}$. $\endgroup$ – Jack M Feb 23 '14 at 13:25
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You probably need to consider that $(f(x))^n$ is another function $g(x)$ which has some properties, some of them coming from the properties of function $f(x)$. This could be very useful to know for the analysis of the behavior of function $g(x)$ (derivatives, zero's, Taylor series expansion,..). But, for example, with regard to integration, I do not see (at first glance) what we could use this relation for, at least in a very general manner.

The above blabla applies if the exponent is a number. The problem starts to be very different if you consider $h(x)=f(x)^{g(x)}$. You will still have very interesting properties of $h(x)$ knowing the properties of $f(x)$ and $g(x)$.

For illustration purposes, I suggest you plot on the same graph $x^3$ and $x^4$ (say for $0<x<2$; the curve will look "similar". But also plot $cos(x)$ and $cos(x)^{cos(x)}$ (say for $0<x<\pi/2)$; these are very different.

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The best approach to plot $h(x)=f(x)^{g(x)}$ is to take and plot the logarithms $$\log(h(x))=g(x)*\log(f(x))$$ like we were doing by hand "many years ago" using log paper.
Nowadays, with a good computer program, you do not have such a limitation.

Of course to remain in the reals, $f(x)$ shall be positive, at least in the $x$ range of interest.
So you cannot make a real values plot of $cos(x)^{sin(x)}$, unless for $\pi/2 \le x \le \pi/2$.
Otherwise, if you want to go to complex values, the you shall split into two plots for the Real and Imaginary , or for the Modulus and Phase, components.

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Why not just plot it using free software (e.g., WolframAlpha)?

enter image description here

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