Question

Question #befb7

Answer 1

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`