Q.Graph the functions $f(x)=x^3$ and $g(x)=\sqrt[3]{x}$ on the same coordinate plane. Find $f\circ g$ and graph it on the plane as well. Explain your results.

Rather than plotting a complicated curve point by point, recognise it as a reflection, translation, or dilation of an already-known simpler curve. Reflection and translation keep the graph congruent (same size and shape); dilation changes the shape by stretching/compressing.

Reflection (mirror image about a line):

Translation (slide without changing shape), for $c,d>0$:

Dilation (stretch/compress): multiplying $f$ by a constant $k>0$ scales the graph vertically -- $k>1$ stretches it away from the $x$-axis, $0<k<1$ compresses it toward the $x$-axis. Replacing $x$ by $kx$ inside $f$ instead rescales horizontally (e.g. $y=\sin2x$ compresses $y=\sin x$ horizontally by factor 2, halving the distance between successive $x$-intercepts while keeping the same $\pm1$ amplitude).

Combining transformations, in order. To build $y=2\sin(x-1)+3$ from $y=\sin x$: shift right by 1 (translation) $\to y=\sin(x-1)$; stretch vertically by 2 (dilation) $\to y=2\sin(x-1)$; shift up by 3 (translation) $\to y=2\sin(x-1)+3$. The same layered approach handles any $y=k\!\cdot\! f(x\pm c)\pm d$ built from a known base curve $f$ (lines, $|x|$, $x^2,\ x^3,\ x^{1/3},\ \sin x$, etc.), always working from the innermost change (horizontal shift/reflection inside $f$'s argument) outward (vertical stretch, then vertical shift).