Quick questions on Combining functions: sums, differences, products, quotients, squares and reciprocals

For $y = f(x) + g(x)$, add the heights of the two graphs at each $x$. Useful checks:

For $y = f(x) g(x)$, multiply the heights:

The graph of $y = \frac{1}{f(x)}$ comes from $y = f(x)$ by these rules:

$y = \csc x = \frac{1}{\sin x}$ has vertical asymptotes at $x = k \pi$ (zeros of $\sin x$), agrees with $\sin x$ at $\sin x = \pm 1$ (so at $x = \frac{\pi}{2} + k \pi$), and is positive on intervals where $\sin x > 0$.

$y = \sin^2 x$ has zeros at $x = k \pi$ (double roots), touches the $x$-axis there, is bounded above by $1$, and equals $1$ at $x = \frac{\pi}{2} + k \pi$. Its period is $\pi$, half that of $\sin x$.

$y = \frac{\sin x}{x}$ has a hole at $x = 0$ (because both numerator and denominator are zero there) with limit $1$. Elsewhere it inherits zeros from $\sin x$ at $x = \pm \pi, \pm 2 \pi, \dots$ and decays in amplitude like $\frac{1}{|x|}$ as $|x| \to \infty$.

$y = e^{-x} \sin x$ for $x \ge 0$ oscillates with the same zeros as $\sin x$ (at $x = k \pi$), but the amplitude decays. The envelopes are $y = \pm e^{-x}$, and the graph touches them where $\sin x = \pm 1$.

$y = x + \sin x$ has the line $y = x$ as a "spine" with small oscillations of amplitude $1$ added. The graph is always within $1$ of $y = x$ and crosses the line at every multiple of $\pi$.

$\frac{1}{f}$ is not a reflection. The shape distorts: large values become small and vice versa.

For $\frac{f}{g}$, vertical asymptotes come from $g(x) = 0$, not $f(x) = 0$. Zeros come from $f(x) = 0$ (with $g(x) \neq 0$).

$f^2$ has zeros wherever $f$ does. The graph touches the $x$-axis at each one rather than crossing.

For $y = f g$ with $|g| \le 1$, the envelope is $y = \pm f$, not $y = f$ alone. Both branches matter.

If $f$ and $g$ both vanish at the same point with a common factor, you get a hole, not an asymptote. Factor and cancel before concluding.