Question

How do you determine if the equation `f(x) = 2(1/2)^x` represents exponential growth or decay?

Answer 1

`f(x)` is decay

Explanation:

`color(brown)("Moving to the right on the number line")`

As `x` becomes ever greater then `(1/2)^x` becomes less and less but still positive.

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`color(brown)("Moving left on the number line")`

When `x<0` we have `2(2/1)^(|x|)`

The more and more negative `x` becomes the greater `2(2/1)^(|x|)`
becomes. However in doing so we are moving further from 0 in the negative direction. Consequently changing this direction so that you move closer to 0 then `2(2/1)^(|x|)` is decreasing
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Thus `2(1/2)^x` is always decreasing as you move to the right.

`lim_(x->oo) 2(1/2)^x ->0`