$2x + 1 = 2$ $ \ \ \ \ -1 = -1$ $2x\ \ \ \ \ = 1$ $\frac 12 (2x) = \frac 12 (1)$ $\ \ \ \ \ x = \frac 12$ |

This conforms with the symbolic result, $x = \frac {C- B} A$ where $A=2, B = 1$, and $C = 2$. So the solution is $ x = \frac {2- 1} 2 = \frac {1} 2.$

The problem can be restated now: To find a value for $x$ where $f(x) = 2$.

This problem and its solution can be visualized with mapping diagrams for the linear functions $f$ treated as a composition of $m$ followed by $s$.

Check the box in the GeoGebra sketch to see a visualization of the solution. |
On the GeoGebra mapping diagram for the composition, find $2$ on the final target axis. by subtracting
$1$ from $2$. This corresponds to going back on the diagram from $2$ on the
final axis to $2-1=1$ on the middle axis, reversing the arrow that would go from $1$ to $2$ visualizing $s$.Undo $s$ Thus if $m(x) = 1$ then $f(x) = s(m(x))=s(1)=2$ by multiplying $1$ by $\frac 12$. This
corresponds to going back on
the diagram from $1$ on the middle axis to $\frac 12$ on the source
axis, reversing the arrow that would go from $\frac 12$ to $1$ visualizing $m$.Undo $m$ Thus when $x=\frac 12$, $f(x) = s(m(\frac 12))=s(1)=2$ On the GeoGebra sketch you can change the values of $m=A$ and $b=B$ (with the sliders) and $C$ (currently $2$ on the target axis) to see other linear equations $Ax + B = C$ are solved visually with the mapping diagram. |