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 you trace the graph with your mouse pointer, you would be able to observe that the graph is moving upward as you go from *a* to *b, *moving downward from *b* to *c*, and go 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 increasing from *a* to *b *where *b* > *a.*

- If we trace the graph of
*f*with a pencil from*a*to*b*, the trace goes upward. - As
*x*increases (as we go to the right on the*x*-axis),*y*increases (the corresponding*y*value increases ). - If we take two points with coordinates (
*x*_{1},*y*_{1}) and (*x*_{2},*y*_{2}), where*x*_{1},*x*_{2 }are between*a*and*b*, and*x*_{2}is at the right of*x*_{1}, then*y*_{2 }is above*y*_{1.} - If we take two points with coordinates (
*x*_{1},*y*_{1}) and (*x*_{2},*y*_{2}),*x*_{2}and*x*_{1 }between*a*and*b*, if*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?

In addition, there are functions that are increasing (or decreasing) all through out and not just in intervals. For example, The function *f*(*x*) = *e*^{x}* *is increasing all through out*. *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.