Variations and composition: the case of common functions

Key reference functions

There are certain functions whose trends you need to know by heart; these will serve as a foundation for studying more complex functions.

Function	Direction of variation
f(x) = x²	decreasing on ]–∞;0], increasing on [0;+∞[
f(x) = √x	increasing on [0;+∞[
f(x) = 1/x	decreasing on ]–∞;0[ and on ]0;+∞[
f(x) = ax + b, a > 0	increasing on ℝ
f(x) = ax + b, a < 0	decreasing on ℝ

Simple composition rule

When a constant is added or the function is multiplied by a positive number, the direction of variation does not change. For example, g(x) = x² + 3 has the same trend as x².

When multiplying by a negative number, the direction of variation is reversed. For example, h(x) = -3 x √x... Put more simply: if f is increasing on I and k < 0, then k * f is decreasing on I.

Applied example

Let’s examine g(x) = -2x² + 1 on R.

We start with x²: decreasing on ]–∞; 0], increasing on [0; +∞[.

We multiply by –2 (a negative number): the direction of variation reverses everywhere.

Therefore, g is increasing on ]–∞; 0] and decreasing on [0; +∞[. We can verify this using the critical point: g(0) = 1, which is indeed a maximum here (and not a minimum, as is the case for x² alone).

Pitfall to avoid

Do not confuse ‘multiplying the function by a negative number’ (which reverses the direction of variation) with ‘replacing x with -x’ (which reverses the interval, a different kind of symmetry). These two transformations are distinct and yield different results depending on the original function. Always check your reasoning against the definition or the rate of increase if in doubt.