# What is the difference between exponential growth and decay?

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

1. exponential growth for $A<0, k>0,$ and
2. 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 $x\rightarrow\infty,$ $f(x)\approx Ae^{-kx}+C$ 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 $f(x)=Ae^{kx}$ 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.

I'm no mathematician, and there may be more concrete definitions, but this is how I think of a function as exponential "growth" and "decay."

## Theory

If an exponential function is "skyrocketing" (for lack of better terminology) and heads towards $\pm\infty$, 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 $\pm\infty$ (i.e. has some horizontal asymptote), then it is "decaying."

## Concrete

Is $f(x) = -e^x$ exponential growth, decay, or neither?

You can see that as $x$ increases, $f(x)$ will head to $-\infty$. 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 $g(x) = -\frac{1}{e^x}$. You can see that as $x$ increases, $g(x)$ will head to $0$. So, I'd call this a decay.

One simple way to look at this is to compare $e^x$ and $e^{-x}$. The first is "growing" and the second is "decaying." It all depends on the sign of $x$.

Check it out for yourself:

https://www.wolframalpha.com/input/?i=plot+e%5Ex

https://www.wolframalpha.com/input/?i=plot+e%5E-x

They are the same graph except flipped around the y-axis.

Different fields of study will assign different terminology to describe this universally descriptive function. Growth and decay are just particular values for the $A$ and $k$ parameters of the more general function of $f(x)=Ae^{kx}$. There are, of course, even more general forms of the exponential function that have been shown useful to explain a variety of phenomenon. You might be interested in the logistic function:

https://en.wikipedia.org/wiki/Generalised_logistic_function

The logistic function is interesting because it can take on many forms, thresholds and inflections. I think of logistic functions as tamed exponentials. They are also easily differentiable which is useful in areas like neuron networks. They are common in models where you want to take an infinite domain and project it onto a binary response.