Increasing and Decreasing Functions
Big Idea
When we study functions, one of the most important questions we can ask is: "Is the function going up or going down?" This concept — called monotonicity — helps us understand the behavior of functions and is foundational for calculus and beyond.
📈 Definition: Increasing Function
A function $f$ is called increasing on a set $X$ if for any two values of the argument $x_1$ and $x_2$ from the set $X$ such that:
In plain language: As you move to the right along the $x$-axis, the function values go up.
We write: $f(x) \nearrow$ on the set $X$.
📉 Definition: Decreasing Function
A function $f$ is called decreasing on a set $X$ if for any two values of the argument $x_1$ and $x_2$ from the set $X$ such that:
In plain language: As you move to the right along the $x$-axis, the function values go down.
We write: $f(x) \searrow$ on the set $X$.