Understanding Decreasing and Increasing Functions

In this post, we are going to discuss the meaning of increasing and decreasing functions.  Consider the graph of the function f shown below. If we trace the graph with the mouse pointer, we would be able to observe that the trace is moving upward from a to b, moving downward from b to c, and moving upward again from c to d.

In mathematics, we say that the function is increasing from a to b, decreasing from b to c, and increasing again from c to d. Now, let us elaborate what do we mean by increasing functions.

Suppose a function f is strictly increasing (increasing everywhere).

• As x increases (as we move to the right), y also increases (we also move up)
• If we take any two points with coordinates (x₁, y₁) and (x₂,y₂)  and x₂ is to the right of x₁, then y₂ is above y_1.
• If we take any two points with coordinates (x₁, y₁) and (x₂,y₂), and x₂ > x₁, then f(x₂) > f(x₁).

Notice that if we replace the words in orange texts above with their opposite, we will have descriptions of decreasing functions. Can you see why?

The function f(x) = e^x is an example of a function that is strictly increasing. This means that we can take two points anywhere on the graph, and the four descriptions above hold. As an exercise, give an example of a function that decreases all through out.