Question #befb7

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Question #befb7

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Answer 1 · 2017-11-30T20:09:00

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 $g (x)$ $color(green)(g(x))$ , has a constant of proportionality $c$ $color(magenta)c$ if
$XXX g (x) = c \cdot x$ $color(white)("XXX")color(green)(g(x))=color(magenta)c * x$ for all values of $x$ $x$

If $g (x) = 4^{x}$ $color(green)(g(color(red)x)=4^color(red)x$ is a proportional relation then
for $x = 1$ $color(red)x = color(red)1$
we have
$XXX g (1) = 4^{1} = 4$ $color(white)("XXX")color(green)(g(color(red)1))=4^color(red)1=4$
and
$XXX g (1) = c \cdot 1 = 1$ $color(white)("XXX")color(green)(g(color(red)1)=color(magenta)c * color(red)1=color(magenta)1$
which implies $c = 1$ $color(magenta)c=color(magenta)1$

But
for $x = 2$ $color(red)x = color(red)2$
we have
$XXX g (1) = 4^{2} = 16$ $color(white)("XXX")color(green)(g(color(red)1))=4^color(red)2=16$
and
$XXX g (2) = c \cdot 2$ $color(white)("XXX")color(green)(g(color(red)2)=color(magenta)c * color(red)2$
which implies $2 c = 16 xx \to xx c = 8$ $2color(magenta)c=color(magenta)16color(white)("xx")rarrcolor(white)("xx")color(magenta)c=color(magenta)8$

The constant of proportionality, $c$ $color(magenta)c$ , can not be both $1$ $color(magenta)1$ and $8$ $color(magenta)8$

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