**Example LF.1.3 **:$f(x)
=2x$. "Scalar Multiple": $2$.

The arrows in the mapping diagram all end at the a point, $2x$,
twice as far from the position $0$ as the position of $x$ at
the tail of the arrow .

**Motion (Rate) Interpretation:** If $x$ is time and $f(x)$ is
the position of an object on the target axis at time $x$, then at
time $0$ the object is at position $0$. The position at time $x$ is
is precisely $2x$ units greater (less) then $0$ when $x>0$
($x<0$). The object is moving at a velocity of $\frac {\text {2
units of position}} {\text {unit time}} = 2 \frac {\text
{units of position}} {\text {unit time}} $.

**Magnification Interpretation:** If $\Delta x = |x_1-x_0|$ is
the distance between the points on the domain axis
corresponding to numbers $x_0$ and $x_1$, then the distance between
the points on the target axis corresponding to numbers $f(x_0) =
2x_0$ and $f(x_1) = 2x_1$ is

**$\Delta y = |f(x_1)-f(x_0)| = |2x_1-2x_0| =2
|x_1-x_0| = 2 \Delta x$. **

Thus the interval between the points $x_1$ and $x_0$ is magnified by
a factor of $2$.

**Slope Interpretation: **This interpretation is attached
to the graph of $f$ in the coordinate plane. The slope is the $\frac
{\text {rise}}

{\text {run}} =\frac {\Delta y} {\Delta x}= 2$.