How can you sketch the graph of a power of a function?

Question

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 $(cosx)^{sinx}$?

Answer 1

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.

Answer 2

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.