Rational functions.

Quotient of two polynomial functions p(x)/q(x) (where q is not a factor of p), or call them N(x)/D(x) for numerator and denominator polynomials.
Glance below for some examples.
The denominator q, or D, can not be a factor of p, or N, because otherwise this would be a polynomial.

Domain of rational function is R minus the values that make q(x) denominator zero.

x-intercepts are the zeros of the numerator p(x), i.e. p(x)=0 Solve.

y-intercept: set x to 0, if possible.

Vertical asymptotes at x's that make the simplified (i.e. after all cancellations between numerator and denominator) denominator 0. n-degree denominator means ≤n vertical asymptotes: [0..n]. Note: non-rational functions can have infinite vertical asymptotes, e.g. tan

Horizontal asymptote: 0 or 1.
Ratio of dominant/leading terms of numerator and denominator:
Degrees of N and D are the same:
  axn/bxn --> y=a/b
  xn/xn --> y=1 Special case when a=b=1.
Degree of N is greater than degree of D:
  xi/xj   i>j --> none.
  Except see below for oblique asymptote if degree of N is one more than degree of D.
Degree of N is less than degree of D:
  xi/xj   i<j --> y=0

Graph can cross horizontal or oblique asymptote but not vertical asymptotes.
Horizontal asymptote is the limit as x approaches ±∞.
Note: non-rational functions can have 2 horizontal asymptotes:
   arctan has at ±π/2.
   x/√(x2+1) has at ±1
   CDF of standard normal function has at 0 and 1

Examples:
Simplest: y=1/x
           
        

Open-top box with square base x, height y of given volume V.
Surface area (of outside or inside, not both) is S(x)=(x3+4V)/x

Cylinder of radius r and volume V.
Surface area: SA(r)=(2πr3+2V)/r

Oblique/slant asymptote: 0 or 1.
If degree of numerator is one more than degree of denominator:
xi+1/xi
Only one of horizontal and oblique asymptotes are possible.
Divide N by D to get a linear quotient which is the oblique line, ignore remainder.