Direction of variation and symmetry

A decreasing function on each interval

The inverse function f(x) = 1/x is decreasing on ]–∞; 0[ and decreasing on ]0; +∞[.

Note: it is NOT decreasing over the entire R*, as there is a discontinuity at 0. For example, f(-1) = -1 and f(1) = 1: values on either side of 0 cannot be compared directly.

Comparison rule

For all non-zero a and b of the same sign, if a < b then 1/a ≥ 1/b (the inequality reverses when we take the inverse, as the function is decreasing on each interval).

Example: if 2 ≤ x ≤ 5, then 1/5 ≤ 1/x ≤ 1/2.

An odd function

f is odd: for any non-zero x, f(-x) = -f(x). Verification:

f(-x) = 1/(-x) = -(1/x) = -f(x)

This means that the graph is symmetric about the origin O of the coordinate system (central symmetry, or a 180-degree rotation).

Graphical consequence

If we know the graph for $x > 0$, we can obtain the graph for $x < 0$ using this central symmetry with centre O, without needing to recalculate any values.

Common pitfall

Many pupils mistakenly believe that f is decreasing over the entire set R*, and conclude, for example, that 1/(-2) < 1/(3). This is incorrect: one must always check that the two numbers being compared have the same sign before applying the decreasing rule to an interval.