Example OW.FDPC.1 : A function with a single exception in its definition.
Suppose $f(1) = 3$ 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 $0$, the arrow points to $f (0)= 3$ visualizing the "Y-intercept" on the graph of $f$.  Notice also the "hole"  on the graph at $(0,1)$ which is the "Y-intercept" for the linear function $g(x) = 2x+1$.

The function is  increasing for all $x$ in the intervals $(-\infty, 0]$ and  $(0,\infty)$, but not for all real numbers because $f(0) = 3 > f(\frac 1 2) =2$.
You can check this by moving $x$ on the mapping diagram.
You can change the function and the value of $a$ and $f(a)$ to explore this further.

One interesting function to explore with $f(1) = 3$ has $f(x) = \frac {x^2 -1} {x-1}$ otherwise.