Increasing and Decreasing Functions
|Definition: A function $f$ is said to be Increasing on an interval $I$ if for all $m, n \in I$ when $m < n$ we have that $f(m) < f(n)$. A function $f$ is said to be Decreasing on an interval $I$ if for all $m, n \in I$ when $m < n$ we have that $f(m) > f(n)$.|
We should note that $I$ is an interval of the domain of $f$, though, if $I = D(f)$, then $f$ is said to be either always increasing or decreasing. Consider the following graph of a function:
The teal indicates the intervals to where the function is decreasing, while the blue indicates interval to where the function is increasing.
Determining whether a function is increasing or decreasing on a certain interval can be difficult if we don't have an idea of what the graph of $f$ looks like. For example, consider the function $f(x) = x^2 - \ln x$ defined on $(0, \infty)$ and suppose that we want to determine the interval (or intervals) for which this function is increasing. Then we want to find a set for which if $m < n$ then $f(m) < f(n)$. Suppose that $m < n$. Then:(1)
Note that since $m < n$ we have that $m^2 < n^2$. So $n^2 - m^2 > 0$. Furthermore:(2)
From above we see that $f(n) - f(m) > 0$. So $f(m) < f(n)$ and $f$ is increasing on $(0, \infty)$.