Question

Characterization of exponential functions. I just don't understand this concept.

Answer 1

Characterization of exponential functions. If for equal increments of a of the argument x, the values of the function f(x) change by the same ratio, that is f(x + a)/f(x) = b^(a) is constant (independent of x, but dependent on a),

then the function is exponential with: P = f(0) as initial value a = f(1)/f(0) as base f(x) = Pa^(x) ^ I do not understand how they came up with f(x +a)/f(x) = b^(a)

Answer 2

This is the concept above.

Answer 3

sort of odd way to put it

Answer 4

saying that your increase/ decrease is proportional

Answer 5

if you think about it backwards it makes sense. suppose you have an exponential function that looks like say
\[f(x)=50\times (.85)^{\frac{x}{4}}\]
then
\[f(x+a)=50\times (.85)^{\frac{x+a}{4}}=50\times (.85)^{\frac{x}{4}}\times (.85)^{\frac{a}{4}}\]
so
\[\frac{f(x+a)}{f(x)}=(.85)^{\frac{a}{4}}\]

Answer 6

a constant, independent of x

Answer 7

So what does this mean, in english?

Answer 8

How come you used xz for the variable in the exponent, when the function is f(x)