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3

The working definition I have in my head doesn't fit the more rigorous definitions others have put in their answers. I think of exponential growth and decay as being constant percentage growth or decay from or toward an asymptote. My favorite example is temperature of an object, which is shifted with the ambient temperature being the asymptote. I use y = a*b^...


0

Now, I've visited many sites and they all seem to conclude that the following is the definition of an exponential function: $f(x)=ab^x$, $f(x)=ab^{cx+d}$ with suitable restrictions on constants $a,b,c,d$. These definitions are not good (unless the restrictions are $a=1$ in the first case and $ab^d=1$ in the second). A reasonable definition of "...


6

To start with an opinion, I think that this classification exercise is kind of silly. The student is being asked to put functions into some categories without having a clear idea about what those categories mean or are used for. We introduce definitions and categorizations in order to help us understand abstract ideas. A definition without the underlying ...


3

I say the key descriptor of a exponential function is constant multiplicative rate of change, much as the descriptor of a linear function is constant additive rate of change. The function $f(x)=a(1.5)^x$ increases by 50% when $x$ increases by 1: $$\frac{f(x+1)}{f(x)} = \frac{a(1.5)^{x+1}}{a(1.5)^x} = 1.5$$ But adding a non-zero constant changes that: $$\frac{...


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