Rational and modulus functions

A rational function has the form f(x)/g(x), where f and g are polynomial functions, and it is defined only where g(x) ≠ 0.

f(x) = 1/x is defined on R − {0}, and its range is also R − {0}: every non-zero y comes from x = 1/y, while 0 is never an output.

The graph of 1/x has two separate branches, one in the first quadrant and one in the third. As x grows the values shrink towards 0 without reaching it, and as x approaches 0 they grow without bound: 1/0.25 = 4, 1/2 = 0.5, 1/(−0.5) = −2.

The modulus function f : R → R, f(x) = |x|, equals x when x ≥ 0 and −x when x < 0. So |5| = 5, |0| = 0 and |−5| = 5.

The domain of |x| is R and its range is [0, ∞). Its graph is V-shaped with its vertex at the origin: the line y = x for x ≥ 0 and y = −x for x < 0.

|x| measures the distance of x from 0 on the number line, and |a − b| the distance between a and b.