Example 1: $m  = 2; b = 1: f (x) = 2x + 1$

Draw a mapping diagram and graph with a visualization for $f$ on the graph and a mapping diagram for $x=0$ and $x_1=1$ or use the diagram created with GeoGebra to explore further.

Mapping Diagram for $ f (x) = 2x + 1$
Graph for $ f (x) = 2x + 1$

On the GeoGebra figure you can move either $x$ or $x_1$ on the mapping diagram.
On the mapping diagram, given any two distinct points / numbers, $x$ and $x_1$, in the domain, the two corresponding arrows will not intersect between the two axes.
On the graph, if $x<x_1$, then the point $(x,f(x))$ will be lower and to the left of $(x_1,f(x_1))$.
Both these visual features are indicative of a increasing function.