Is the function $y = a^x + b$ exponential?

Question

What exactly is an exponential function? Some of the sources at which I looked said that it's a function where the rate of change at $x$ $(f'(x))$ is proportional to the value at that point $(f(x))$, and others wrote that it is simply a function with $a^x$ in it, where $a$ is a fixed base. So, my question is: if $f(x) = a^x + b$, is $f(x)$ exponential? This function is exponential by the 2nd definition, but not by the first.

Answer 1

This function is exponential by the 2nd definition, but not by the first.

The second definition ("a function with $a^x$ in it") is mathematically vague and makes little sense: think for example of $\sin (a^x)$ - or even $\log(a^x)$

Strictly speaking, the exponential function is one: $f(x)= e^x$

One can extend that to functions of the form $f(x)=a e^x$ or equivalently $f(x)=b^x$ ($b>0$)- and we could call that the "exponential family" of functions. And, loosely speaking, we could call a function that belongs to that family an "exponential function". And we could also define that family by the differential equation $f'(x)=a f(x)$. That would be your first definition.

One can generalize even further, and call "exponential function" any function that has "exponential growth", so that, eg $f(x)/a^x \to 1$. Only in this broad sense your function is "exponential"

Answer 2

I'ts not "exponential" in the sense of the derivative being proportional to the value, no.

It does, however, have "exponential growth", in the sense that there's a constant $C$ with $$ |f(x)| \ge C u^x $$ for large enough $x$ and for some $u > 1$.

In computer science, such functions are sometimes sloppily called 'exponential', even though they could be super-exponential (e.g., $f(x) = 2^{2^x}$). In CS, functions of a variable $x$ often represent how much computation is needed to solve some problem of size $x$, and "exponential" means "at least as bad as a true exponential function" in the sense that solving this problem will require time greater than or equal to exponential time. In short, you'll sometimes see "exponential" used very informally to mean "having at least exponential growth".