In the study of calculus, understanding how functions behave is crucial. One key aspect of this is identifying when a function is increasing or decreasing.
Defining Increasing and Decreasing Functions
A function $f(x)$ is said to be increasing on an interval if, for any two points $x_1$ and $x_2$ in that interval where $x_1 < x_2$, we have $f(x_1) < f(x_2)$. Conversely, a function is decreasing on an interval if, for any two points $x_1$ and $x_2$ in that interval where $x_1 < x_2$, we have $f(x_1) > f(x_2)$.
Note
It's important to distinguish between strictly increasing/decreasing and non-strictly increasing/decreasing functions. The definitions above describe strictly increasing/decreasing functions. For non-strict versions, we would use $\leq$ and $\geq$ instead of $<$ and $>$.
The Role of Derivatives
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Note
Introduction to Increasing and Decreasing Functions
Understanding how functions behave is a fundamental aspect of calculus. One of the key behaviors we study is whether a function is increasing or decreasing on a given interval.
A function is increasing on an interval if, as you move from left to right, the function values go up.
A function is decreasing on an interval if the function values go down as you move from left to right.
AnalogyThink of an increasing function like walking uphill and a decreasing function like walking downhill.
DefinitionIncreasing Function
A function f(x) is increasing on an interval if for any two points x1<x2 in that interval, f(x1)<f(x2).
DefinitionDecreasing Function
A function f(x) is decreasing on an interval if for any two points x1<x2 in that interval, f(x1)>f(x2).
