Question #befb7

1 Answer
Nov 30, 2017

Answer:

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

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

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

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