Geometry of Functions I: Increasing and Decreasing Functions

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Lesson Summary: 
We provide the intuition and the rigorous mathematical definition of increasing and decreasing functions.
Lesson Inputs: 
Functions
Open and Closed Intervals

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Lesson: 

Let's jump right into the intuition of increasing and decreasing functions with pictures. An increasing function is one where as we go from left to right on the graph the function should be getting higher. The figure below illustrates this.

The Intuition of an Increasing Function

A decreasing function is one where as we go from left to right on the graph the function should be getting lower. Again, the figure below illustrates this.

The Intuition of an Decreasing Function

Typically when we talk about a function increasing or decreasing, we don't mean the function over the whole domain. The function will increase or decrease over some piece of the domain of the function. The picture below shows a function increasing in some interval on the $x$-axis, then decreasing over another, and then finally increasing over a third interval.

Function Increasing and Decreasing In Intervals

Now that we have the intuition of increasing and decreasing functions, let's look at the exact definition of increasing and decreasing functions. We denote $I$ as an open or closed interval. When we say that $f(x)$ increases on $I$ we mean the following:

Increasing Function

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So the idea is that we can pick any two <Sign in to see all the formulas> in $I$ and should find that <Sign in to see all the formulas> when the function is increasing. See the figure below.

The Definition of an Increasing Function

Similarly for decreasing functions, when we say that $f(x)$ decreases on $I$ we mean the following:

Decreasing Function

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Finally, we provide the following two problems as examples of determining increasing and decreasing functions: