Domain, Range, and the Standard Properties

Domain $=$ set of $x$-values the graph covers; range $=$ set of $y$-values it reaches. A filled dot / solid edge includes the endpoint (closed); an open dot / asymptote excludes it (open). Reading them off a sketch is often faster than algebra for standard curves.

• Even root $\sqrt[\text{even}]{g(x)}$: need $g(x)\ge 0$.
• Denominator: need denominator $\neq 0$.
• Logarithm $\log g(x)$: need $g(x)>0$; base must be $>0,\neq1$.

When several appear, the domain is the intersection of all the individual conditions.

Common techniques:

• Solve for $x$: rearrange $y=f(x)$ to $x=\dots$; the range is the $y$ for which $x$ is real/in-domain.
• Quadratic on an interval: check the vertex and the endpoints; mind whether endpoints are included.
• $a\sin x+b\cos x+c$ lies in $[c-\sqrt{a^2+b^2},\,c+\sqrt{a^2+b^2}]$.
• AM–GM for a reciprocal pair: $t+\dfrac1t\ge2$ for $t>0$ (and $\le-2$ for $t<0$), so e.g. $\dfrac{x^2+1}{x}$ never lands in $(-2,2)$, and $\dfrac{x}{x^2+1}\in\left[-\tfrac12,\tfrac12\right]$.
• Linear-fractional $\dfrac{ax+b}{cx+d}$ takes every real value EXCEPT $\dfrac ac$ (the horizontal asymptote) — the one value a "which is not in the range?" option list is fishing for.
• A trig substitution for a cubic pattern: on $[-1,1]$, $x=\sin\theta$ turns $x(4x^2-3)=4\sin^3\theta-3\sin\theta=-\sin3\theta$, and $x=\cos\theta$ turns $4x^3-3x$ into $\cos3\theta$ — so the range is $[-1,1]$ immediately.

• Even: $f(-x)=f(x)$ for all $x$ (e.g. $x^2,\ \cos x,\ |x|$).
• Odd: $f(-x)=-f(x)$ for all $x$ (e.g. $x^3,\ \sin x,\ x$). An odd function defined at 0 has $f(0)=0$.
• Test by computing $f(-x)$ and comparing. If it matches neither, the function is neither.
• The rationalisation that recurs: $\big(x+\sqrt{1+x^2}\big)\big(-x+\sqrt{1+x^2}\big)=1$, so $\ln\!\big(-x+\sqrt{1+x^2}\big)=-\ln\!\big(x+\sqrt{1+x^2}\big)$: the function $\ln\!\big(x+\sqrt{1+x^2}\big)$ (which is $\sinh^{-1}x$) is odd. The same trick shows $\log\dfrac{1-x}{1+x}$ is odd and $\dfrac{a^x-1}{a^x+1}$ is odd, while $x\cdot\dfrac{a^x-1}{a^x+1}$ is even.

$f$ is periodic with period $T>0$ if $f(x+T)=f(x)$ for all $x$; the smallest such $T$ is the period.

• $\sin x,\cos x$: period $2\pi$; $\tan x$: period $\pi$.
• $\sin(kx)$ has period $\tfrac{2\pi}{|k|}$; $\sin^2 x$ has period $\pi$.
• Period of a sum is the LCM of the individual periods.

$|x|=x$ for $x\ge0$ and $-x$ for $x<0$; it is even, with range $[0,\infty)$.

• $x+|x|=0$ for $x<0$ and $2x$ for $x\ge0$ — range $[0,\infty)$.
• $|x-a|+|x-b|$ (with $a<b$) has minimum value $b-a$, attained for all $x\in[a,b]$.

• Reciprocal $\dfrac{1}{x-a}$: domain $x\neq a$, range $y\neq0$; vertical asymptote $x=a$, horizontal $y=0$.
• Sign $\dfrac{x}{|x|}$: equals $+1$ for $x>0$, $-1$ for $x<0$; undefined at 0.
• Exponential $a^x\,(a>0)$: domain $\mathbb{R}$, range $(0,\infty)$, continuous and differentiable everywhere.