7

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 ...


4

If the domain was the real numbers, I would say that it raises two to the power of its input. If the domain was positive integers, I would break your model and say that it multiplied together $x$ copies of $2$ because I think it is more intuitive to describe what exponentiation actually does that to imply that it is as natural as addition and multiplication.


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^...


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{...


2

"The function $h$ raises $2$ to the $x$, where $x$ is the input" and "The function $h$ raises $2$ to the $x$th power, where $x$ is the input" both ape the structure of $f$ and $g$ and, despite containing a clause, sound less awkward/contrived than the alternatives.


2

One example for "good" exponential growth is cryptographics. Linear key length growth vs. exponential growth for the effort to break the key. In this example we are kept safe by the exponential growth.


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