Use the derivative to determine whether the function `y = 8e^x - 12 ` is strictly monotonic on its entire domain and therefore has an inverse function.

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We are asked to determine if the function ` y=8e^x-12 ` has an inverse function by finding if the function is strictly monotonic on its entire domain using the derivative. The domain is all real numbers.

`y'=8e^x ` and y'>0 for all real x so the function is strictly monotonic (in this case strictly increasing) on its entire domain and thus has an inverse function.

The graph of the original function, which we can see is always increasing:

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