SL 5.2—Increasing and decreasing functions

Before diving into calculus tools, it helps to have a clear picture of what "increasing" and "decreasing" actually mean mathematically.

Increasing Function: A function $f(x)$ is increasing on an interval if, for any two points $x_1$ and $x_2$ in that interval with $x_1<x_2$, we have $f(x_1)<f(x_2)$. In other words, as $x$ moves to the right, the function values go up.

Decreasing Function: A function $f(x)$ is decreasing on an interval if, for any two points $x_1$ and $x_2$ in that interval with $x_1<x_2$, we have $f(x_1)>f(x_2)$. As $x$ moves to the right, the function values go down.

Think of it this way: if you walk along the graph from left to right, an increasing function takes you uphill, and a decreasing function takes you downhill.