Functions

Every output comes from at most one input. Horizontal-line test: ≤ 1 intersection. Required for an inverse.

At least two different inputs share an output. Fails the horizontal-line test.

Every element of the stated co-domain is achieved as an output. Range = co-domain.

Restrict a many-to-one function to ONE side of its turning point to force one-to-one. Example: $x^2$ on $\mathbb{R}$ is many-to-one; on $[0,\infty)$ it's one-to-one with inverse $\sqrt{x}$.

Graph symmetric about the $y$-axis. Polynomials: only even powers + constant. Examples: $x^2,\cos x,|x|$.

Graph has $180^\circ$ rotational symmetry about origin. Polynomials: only odd powers, no constant. Examples: $x^3,\sin x,1/x$.

At least one $x$ with $f(-x) \ne \pm f(x)$. Mixed-degree polynomials like $x^2+x$.

Only $f(x)=0$ satisfies both. $f=f$ and $f=-f$ ⇒ $2f=0$ ⇒ $f=0$.

Set the radicand $\ge 0$ and solve. Don't forget to flip the inequality if $a<0$.

$y=c+k\sqrt{ax+b}$ starts at $(x_0, c)$ where $x_0$ is the boundary. If $k>0$ the curve rises; if $k<0$ it falls below $c$.

Bare $\sqrt{\dots}\ge 0$ ⇒ range $\ge 0$. Outer constant shifts the range floor; outer negative gives range $\le c$.

Proper fraction (deg numerator $<$ deg denominator). Factor denominator into distinct linears: $\dfrac{f(x)}{(x-r_1)(x-r_2)} = \dfrac{A}{x-r_1}+\dfrac{B}{x-r_2}$.

Find $A_i$ by covering up $(x-r_i)$ in the original and substituting $x=r_i$ into what remains. Works because every other term has $(x-r_i)$ as a factor and vanishes.

$(x-r)^{2}$ in denominator ⇒ TWO pieces: $\dfrac{A}{x-r} + \dfrac{B}{(x-r)^{2}}$.

Different rules on different intervals. Domain = union of the intervals. Evaluate by selecting the rule for the input's interval.

Inverses require one-to-one. Integration $\int_{-a}^{a}f$ vanishes if $f$ is odd (saves work!). Symmetry gives quick range arguments and graphs faster sketching.

If $f$ is odd and defined at $0$, then $f(0)=0$ (set $x=0$ in $f(-x)=-f(x)$). If $f(0)\ne 0$, $f$ cannot be odd.

Numerator MUST be lower degree than denominator. If not, polynomial-divide first; the quotient is a polynomial part plus a proper fraction.