Definition: We say that a $f$ is
(montonically) increasing for a set D, if whenever
$a < b $ for $a$ and $ b \in D$ , $f(a) < f(b)$.
We say that a $f$ is (montonically) decreasing
if whenever $a < b $ for $a$ and $ b \in D$, $f(a) > f(b)$. Definition: We say that a $f$ is monotonic for a set D, if either $f$ is (monotically) increasing or $f$ is (monotonically) decreasing
for the set $D$.
Graph
Mapping Diagram
Increasing:
If $a < b $ then $f(a) < f(b)$.
Decreasing:
If $a < b $ then $f(a) > f(b)$.
Definition: We say that a $f$ is (monotonically) increasing or decreasing at a number or point $c$, if either $f$ is (monotonically) increasing or $f$ is (monotonically) decreasing
for an open interval $D = (a,b)$ with $c \in D$.
Mapping Diagram Visualization: An arrow (pointing up or down) on the
target axis at the point representing the value of the function at $c$,
$f(c)$, can be used to indicate
whether $f$ is increasing (pointing up) or
decreasing (pointing down) at $c$.
When appropriate the magnitude of such a target axis arrow can indicate
the rate of increase or decrease of the function values.
Example: Consider $y = f(x) = \frac 12 x$. Then $f$ is increasing for $(-\infty, \infty)$. So $f$ is increasing at any number $x$.
Using $\frac 12$ as the rate of increase of the function values, the following mapping diagram visualizes this information.
When $x=2$, the arrow from $x=2$ on the domain axis to $f(x)=f(2)=\frac
12 (2) = 1$ on the target axis shows the function connection.
The
arrow pointing up on the target axis at $f(x)=1$ with length $\frac 12$
indicates that when $x=2$ the function $f$ is increasing at a rate of $\frac 12$
units of $y$ per unit of $x$ .
