Polynomial of Trig Functions

Question

Let $f(x)=\cos(x)$ and $g(x)=x^3-x^2-6x$ Has anyone investigated about the shape of $f(g(x))$? It's like a wave function, but the roots converge as $|x|$ increases.

Answer 1

here is your graph of $f(g(x))$.

what do you want else, your question is not clear!

Answer 2

You can use this Desmos.com link (open in a new tab) to investigate graphs of the form

$$f(x)=\cos(g(x))$$

where

$$ g(x)=ax^3+bx^2+cx+d $$

by changing the values of the coefficients to see the effect on the graph.

Answer 3

The roots of $f$ are

$$\pm\frac\pi2+2k\pi.$$

Obviousy, the function $g$, which is of the cubic type, reaches these values faster and faster for growing $x$, as it is super-linear.

The roots of $f(g(x))$ are obtained by solving the cubic equation

$$g(x)=x^3-x^2-6x=\pm\frac\pi2+2k\pi$$ which lead to a nasty expression that I spare you. For large $x$, you can approximate as

$$x^3\approx\pm\frac\pi2+2k\pi$$ or

$$x\approx\sqrt[3]{\pm\frac\pi2+2k\pi}\propto\sqrt[3]{k}.$$

To get an intuitive understanding of the curve behavior, you can see it as a stretched version of a cosinusoid, by the function $g^{-1}$.