Given the exponential function A(x)=P(1+r)^x, what value for r will make the function a decay function

The function will be a decay function for
.. |1 +r| < 1
.. -1 < 1 +r < 1
.. -2 < r < 0

Any value of r in the interval (-2, 0) will make the function a decay function.

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If x can have non-integer values, then you probably want r ∈ (-1, 0).