Given two functions $f(x),g(x)$ so that $f(x)=-\frac{x^3}{3}+x^2+1,g(x)=5-2x$. Find the ranges of $x$ so that $f(g(x))<g(g(x))$ .

Question

Given two functions $f(x), g(x)$ so that $f(x)= -\dfrac{x^{3}}{3}+ x^{2}+ 1, g(x)= 5- 2x$ . Find the ranges of $x$ so that $$f\left ( g(x) \right )< g\left ( g(x) \right )$$

Is an easy Desmos problem but my teacher hate it https://www.desmos.com/calculator/jjfaugkkku I see that $f\left ( g(x) \right )\gtreqqless g\left ( g(x) \right )$ is related to $f(x)\gtreqqless g(x)$ . So how can I use $g(x)= 5- 2x$ in a smarter way ? Thank you.

Answer 1

You have to find the $y$-intervals where $$f(y)<g(y)\qquad(y\in{\mathbb R})\ .$$ There are two such intervals, say $A$ and $B$; you get them by solving a cubic equation. You then have to find the $x$-intervals for which $$y:=g(x)\in A\cup B\ .$$