Variations of a function

Increasing or decreasing?

Studying the variations of a function means observing how f(x) changes as x increases.

• f is increasing on an interval I if, for all a and b in I such that a <= b, we have f(a) <= f(b) (the images follow the same direction as the antecedents).
• f is decreasing on I if a <= b implies f(a) >= f(b) (the images move in the opposite direction).

Practical example

The ‘square’ function f(x) = x²:

• is decreasing on ]–∞; 0] (the closer x gets to 0 from the left, the lower x² becomes)
• is increasing on [0; +∞[ (the further x moves away from 0 to the right, the higher x² becomes)

The table of variations

We summarise this information in a table:

x	-∞	...	0	...	+∞
f(x)		decreasing (downward arrow)	0	increasing (upward arrow)

A downward arrow indicates a decrease, an upward arrow indicates an increase.

Common pitfall

A function that first decreases and then increases is not ‘contradictory’: this is perfectly normal; a function may change its direction of variation several times over its domain. Be careful not to confuse the sign of f(x) (positive or negative) with the direction of variation (increasing or decreasing): these are two different pieces of information that are not interpreted in the same way.

Minimum and maximum

When a function first decreases and then increases, it has a minimum at the point of change (here 0 for f(x) = x², where the minimum is 0). Conversely, a function that first increases and then decreases has a maximum at that point.