"A colleague came across this terminology question. What are the definitions of exponential growth and exponential decay? In particular: 1) Is f(x)=−e^x exponential growth, decay, or neither? 2) Is g(x)=−e^−x exponential growth, decay, or neither? Consider f(x)=Ae^kx. I can't find any sources that specify A>0. My answer is that: f exhibits 1. exponential growth for A>0,k>0, and 2. exponential decay for A>0,k<0 whereas |f| exhibits 3. exponential growth for A<0,k>0, and 4. exponential decay for A<0,k<0.

sombereki51 2022-09-22 Answered
A colleague came across this terminology question.
What are the definitions of exponential growth and exponential decay? In particular:
1) Is f ( x ) = e x exponential growth, decay, or neither?
2) Is g ( x ) = e x exponential growth, decay, or neither?
Consider f ( x ) = A e k x .. I can't find any sources that specify A>0. My answer is that:
f exhibits
1. exponential growth for A>0,k>0, and
2. exponential decay for A>0,k<0
whereas | f | exhibits
3. exponential growth for A<0,k>0, and
4. exponential decay for A<0,k<0.
In case (3) we shouldn't call f an exponential growth function without noting that it is "negative growth". Also it wouldn't be called it an exponential decay function without specifying the "direction of decay", so it is neither.
In case (4) it's neither as well. One should specify that it is the magnitude of f which decays exponentially although f is increasing in value. Although f is increasing in value, is it growing? It seems odd to say it is exponentially growing.
It just doesn't sit right with me to refer to a function as growing if it is decreasing in value. Certainly, it's magnitude may be growing.
Next consider a function with exponential asymptotic behavior (e.g. logistic) so that as x , f ( x ) A e k x + C for some k>0. I feel the best way to describe this would be "exponential decay towards C" with a qualification as being from as being from above or below depending on the sign of A.
If someone is to just use the terminology "exponential growth (decay)", it implies f ( x ) = A e k x with positive A and k>0 (k<0) unless there is a specific context or further clarification as to what the actual nature of the function is.
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Answers (2)

Gauge Bishop
Answered 2022-09-23 Author has 3 answers
Theory
If an exponential function is "skyrocketing" (for lack of better terminology) and heads towards ± , then it's "growing" (you can think of it as "absolute" growth, and disregard the sign).
If an exponential function is "slowing down" (again, for lack of better terminology), and heads towards something other that ±∞ (i.e. has some horizontal asymptote), then it is "decaying."
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Lustyku8
Answered 2022-09-24 Author has 2 answers
Concrete
Is f ( x ) = e x exponential growth, decay, or neither?
You can see that as x increases, f(x) will head to . So, I'd call this a growth.
Is g ( x ) = e x exponential growth, decay, or neither?
For easier visualization of a corresponding graph, let us rewrite g ( x ) = 1 e x . You can see that as x increases, g(x) will head to 0. So, I'd call this a decay.
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