# 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 (
*x*_{1},*y*_{1}) and (*x*_{2},*y*_{2}) and*x*_{2}is to the right of*x*_{1}, then*y*_{2 }is above*y*_{1.} - If we take any two points with coordinates (
*x*_{1},*y*_{1}) and (*x*_{2},*y*_{2}), and*x*_{2}>*x*_{1}, then*f*(*x*_{2}) >*f*(*x*_{1}).

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.