Geometry of Functions I: Increasing and Decreasing Functions

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.

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.

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.

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:

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.

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

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

• Prove that <Sign in to see all the formulas> decreases on the interval <Sign in to see all the formulas> and increases on the interval <Sign in to see all the formulas>.
• Prove that <Sign in to see all the formulas> increases everywhere.