Decreasing Function

Question 1

What is a decreasing function? (Definition)

Answer

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 $x_1 < x_2 , f(x_1) \geq f(x_2)$

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

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 $] -\infty ; 0 [$

Question 2

How to determine if a function is decreasing?

Answer

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 $f'(x) < 0$ is simplified as $x < 0$ 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.

Question 3

How to determine if a linear/affine function is decreasing?

Answer

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.

Decreasing Function