# Which functions are examples of exponential decay? a) f(x) = (1/3)^x b) f(x) = 3^2x c) f(x) = (1/4)^-x d) f(x) = 4^-x

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Which of the following are exponential decay models:

**Note that if the argument is less than one with a positive exponent, the model is a decay model **

(a) ` ` `f(x)=(1/3)^x` is decay since `0<1/3<1`

(b) `f(x)=(3)^(2x)` is growth since 3>1

(c) `f(x)=(1/4)^(-x)` : rewrite as `f(x)=((1/4)^(-1))^x=4^x` Since 4>1this is a growth model.

(d) `f(x)=4^(-x)` : rewrite as `f(x)=((4^(-1))^x=(1/4)^x` which is a decay since `0<1/4<1`

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(a) and (d) are decay models

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Exponential decay refers to a situation where a quantity decreases at a rate proportional to its value. For a function f(x), f'(x) = -A*f(x)

For the given functions:

- f(x) = `1/3^x` : `f'(x) = -ln(3)/3^x` = `-ln(3)*f(x)` This is an example of exponential decay

- f(x) = `3^(2x)` : f'(x) = `2*ln(3)*f(x)` This is not an example of exponential decay

- f(x) = `(1/4)^-x` = `4^x` : `f'(x) = ln(2)*2*f(x)` This is not an example of exponential decay

- f(x) =` 4^-x` = `1/4^x` : f'(x) = `-ln(2)*2*f(x)` This is an example of exponential decay