Increasing and Decreasing Functions

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:

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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)
\begin{align} \quad f(m) = m^2 - \ln m \quad \mathrm{and} \quad f(n) = n^2 - \ln n \end{align}

Note that since $m < n$ we have that $m^2 < n^2$. So $n^2 - m^2 > 0$. Furthermore:

(2)
\begin{align} \quad f(n) - f(m) &= (n^2 - m^2) - \ln n + \ln m \\ &= (n^2 - m^2) + \ln \frac{m}{n} \end{align}

From above we see that $f(n) - f(m) > 0$. So $f(m) < f(n)$ and $f$ is increasing on $(0, \infty)$.

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