How to grasp such level concepts to solve these questions( lectures or books or any other resource)

One thing I knew after playing on Desmos (place to write equations and see their graph) was this:

$2^x = x^2$

My brain goes and picks values & does trial-error. Work out both sides. Note which is bigger. Every time the answer in the last column flips, they were equal somewhere in between (the line/curve must have intersected).

$x$	$2^x$	$x^2$	bigger
$-3$	$0.125$	$9$	$x^2$
$-2$	$0.25$	$4$	$x^2$
$-1$	$0.5$	$1$	$x^2$
$0$	$1$	$0$	$2^x$ ← flip
$1$	$2$	$1$	$2^x$
$2$	$4$	$4$	equal
$3$	$8$	$9$	$x^2$ ← flip
$4$	$16$	$16$	equal
$5$	$32$	$25$	$2^x$ ← flip
$6$	$64$	$36$	$2^x$
$7$	$128$	$49$	$2^x$

---

How to learn this?

Needed: know the shape of common graphs, place two on top of each other, count crossings.

1. Shapes. $x^2$ is a U. $2^x$ stays flat, then shoots up. Also $1/x$, $\sqrt{x}$, $\log x$. Five shapes. Memorise.
2. Flip rule. Bigger here, smaller there, means equal somewhere between. The table above is the whole method.
3. Growth order. $2^x$ beats $x^2$ in the long run, not at the start. The early loss is why this has three answers.

Resources.

1. Desmos. Type 2^x and x^2. Look. Then 3^x vs x^2. Then 2^x vs x^3.
2. NCERT Class 11 Chapter 2. Graph shapes. Eddie Woo had a fun video on graphs.
3. Habit/Heuristics: if the equation looks impossible, picture both graphs, count crossings. "How many solutions" questions do not want the solutions (so it makes our life a bit easier)