Example OW.FDPC.3:A function with three cases on intervals. [Three
rates.]
Suppose $f(x) = x$ when $x < 0$ ; $f(x) = -x +1$ when $ 0 < x
\le 2$, and $f(x) =1$ otherwise.
Draw a mapping diagram and graph of $f$ yourself, or consider the GeoGebra figures and table below.
Martin Flashman, Dec. 20, 2013,
Created with GeoGebra
Notice how the points on the graph are paired with the points
on the mapping diagram.
Given a point / number, $x$, on the source line, there is a unique
arrow meeting the target line at the point / number, $f(x)$.
which corresponds to the function's value for $x$
Check the box to see more points and arrows
corresponding to data on the table.
When the point $x$ in the domain is $( -\infty,9) $, the arrow
points to $f (x)= x$, while when $x$ is in the interval $[0,2)$ the
arrow points to $ f(x)=-x+1$ on the target axis, and otherwise the
arrow points to $f(x) =1$. Notice also the "steps" on
the graph at $(0,1)$ and $(2,1)$ which are points where the
graph is disconnected.
The function is increasing for the interval
$(-\infty,0)$, but decreasing for the interval $[0,2)$ and constant
on the interval $[2,\infty)$.
You can check this by moving $x$ on the mapping diagram.
You can change the function and the values of $a$ and $b$ where the
"steps" occur to explore this further.