Manually plotting some particular graphs

Question

How to plot graphs like these manually:

1) $f(x)=\ln(1+x^2)$

2) $f(x)=\frac8{2+x^2}$

3) $f(x)=\frac{\sin x}{\sqrt{1+\tan^{2}x}}+\frac{\cos x}{\sqrt{1+\cot^{2}x}}$

I have no idea how to plot the nature of these functions on pen and paper without using any software.

Answer 1

For 1 see the asymptotic $f_1(x) \sim \ln(x^2) = 2\ln |x|$ and note that $1+x^2 \ge 1$ so $f_1(x) \ge 0$ and opens up like $\ln$ to both sides (resembles a smooth bucket)

Similarly, the shape of $f_2$ is like the shape of $\frac1{1+x^2}$, somewhat like a bell with its only maximum at $0$ with value $4$.

For 3 just try some algebraic manipulations: $\sqrt{1+\tan^2 x} = |\sec x| \sqrt{\cos^2 x + \sin^2 x} = |\sec x|$ and analogously $\sqrt{1+\cot^2 x} = |\csc x|$.
This shows $f_3(x) = \sin x|\cos x| + |\sin x|\cos x$. Now using cases you'll get a piecewise function wich is $\pm\sin 2x$ or $0$ depending on the interval of $x$.

Generally it's always good to

• simplify the function as much as possible
• do a curve discussion (roots, extrema, inflection points)
• compare to known graphs