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 $f(x)=A{e}^{kx}.$. 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.

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 $x\to \mathrm{\infty},$ $f(x)\approx A{e}^{-kx}+C$ 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 $f(x)=A{e}^{kx}$ 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.