Example OW.FDPC.2 :A function with two cases on intervals. [Two rates.]
Suppose $f(x) = x$ when $x \ge 0$ and $f(x) = -x +1$ otherwise.
Draw a mapping diagram and graph of $f$ yourself, or consider the GeoGebra figures and table below.

Martin Flashman, Nov. 9, 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 in the domain is $[0,\infty)$, the arrow points to $f (x)= x$, while otherwise the arrow points to $-x+1$ on the target axis.  Notice also the "step"  on the graph at $(0,0)$ which is the "Y-intercept" where there the graph is disconnected.

The function is  increasing for the interval  $(0,\infty)$, but decreasing for the interval $(-\infty,0)$.
You can check this by moving $x$ on the mapping diagram.
You can change the function and the value of $a$ where the "step" occurs to explore this further.
One interesting function to explore has $f(x) =1$ when $x \ge 0$  and  $f(x) = 0$ otherwise.