Decreasing Function

A function $ f $ is strictly decreasing if for any $$ x_1 < x_2, f(x_1) < f(x_2) $$ (signs are inverted)

In other words, $ f $ has a decreasing direction of variation, when $ x $ decreases, $ f(x) $ also decreases (not necessarily by the same quantity).

A function is said to be decreasing (not strictly, in the broad sense) if for all $$ x_1 < x_2, f(x_1) \leq f(x_2) $$

Example: The function $ f(x) = x + 1 $ is decreasing over its whole domain of definition $ \mathbb {R} $

The decrease of a function can also be defined over an interval.

Example: The function $ f(x) = x^2 $ is strictly decreasing over $ \mathbb{R}^+ $ also noted $ x < 0 $ or also $ ] -\infty ; 0 [ $

Several methods allow to to find the direction of variation for knowing if a function is decreasing:

- From its derivative: When the derivative of the function is less than $ 0 $ then the function is decreasing.

Example: The derivative of the function $ f(x) = x^2+1 $ is $ f'(x) = 2x $, the calculation of $ f'(x) > 0 $ gives $ x > 0 $ so the function $ f $ is decreasing when $ x > 0 $

- From its equation : Some functions are notoriously decreasing, ie. the inverse function, the opposite of increasing functions, etc.

Example: $ \frac{1}{x} $ is decreasing over $ \mathbb{R}^* $

- From the curve of the function: a decreasing function has its curve which is directed downwards.

A linear function of the form $ f(x) = ax + b $ is decreasing over $ \mathbb{R} $ when the coefficient $ a $ is positive ($ a < 0 $). If $ a $ is positive then the function is increasing.