Question

How do you determine if the equation `f(x) = 5(1/6)^x` represents exponential growth or decay?

Answer 1

Logic corrected!`" "-> " decay"`

Explanation:

Total rewrite!

Consider the case of: `" "x<0`

We have: `5/((1/6)^x) = (5xx6^x)/(1^x) = 5xx6^x`

Thus as the magnitude of negative `x` increases the value of y also increases.

So as {x: x < 0} increases (moved to the right) it is the reverse of what I previously wrote, so we have decay.

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Consider the case of `" "x=0`

We have: `" " 5(1/6)^0 = 5xx1 = 5`

also we have:

`5(1/6)^(-0)>5(1/6)^0>5(1/6)^(+0)`

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Consider the case of `" " 0 < x`

The denominator of `1/6` increases as `x` increases in `(1/6)^x`
So `(1/6)^x` becomes smaller and smaller as `x` increases.

This means that `5(1/6)^2` decreases as `x` increases.

So again we have decay

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In conclusion: the equation is in decay