SL 5.2—Increasing and decreasing functions

One of the most important questions we can ask about a function is: where is it going up, and where is it going down? This is the study of increasing and decreasing functions , a concept that sits at the heart of curve sketching, optimisation, and real-world modelling.

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.

These are strict definitions , the function value must be strictly greater or strictly less (not equal) as you move right along the interval.