# Question befb7

Nov 30, 2017

Use of the expression "constant of proportionality" implies that the function is a direct proportion relation. This one is not; so there is no "constant of proportionality"

#### Explanation:

A function, this this case $\textcolor{g r e e n}{g \left(x\right)}$, has a constant of proportionality $\textcolor{m a \ge n t a}{c}$ if
$\textcolor{w h i t e}{\text{XXX}} \textcolor{g r e e n}{g \left(x\right)} = \textcolor{m a \ge n t a}{c} \cdot x$ for all values of $x$

If color(green)(g(color(red)x)=4^color(red)x is a proportional relation then
for $\textcolor{red}{x} = \textcolor{red}{1}$
we have
$\textcolor{w h i t e}{\text{XXX}} \textcolor{g r e e n}{g \left(\textcolor{red}{1}\right)} = {4}^{\textcolor{red}{1}} = 4$
and
color(white)("XXX")color(green)(g(color(red)1)=color(magenta)c * color(red)1=color(magenta)1
which implies $\textcolor{m a \ge n t a}{c} = \textcolor{m a \ge n t a}{1}$

But
for $\textcolor{red}{x} = \textcolor{red}{2}$
we have
$\textcolor{w h i t e}{\text{XXX}} \textcolor{g r e e n}{g \left(\textcolor{red}{1}\right)} = {4}^{\textcolor{red}{2}} = 16$
and
color(white)("XXX")color(green)(g(color(red)2)=color(magenta)c * color(red)2#
which implies $2 \textcolor{m a \ge n t a}{c} = \textcolor{m a \ge n t a}{16} \textcolor{w h i t e}{\text{xx")rarrcolor(white)("xx}} \textcolor{m a \ge n t a}{c} = \textcolor{m a \ge n t a}{8}$

The constant of proportionality, $\textcolor{m a \ge n t a}{c}$, can not be both $\textcolor{m a \ge n t a}{1}$ and $\textcolor{m a \ge n t a}{8}$