The Exponential Function and Its Applications

Pure Mathematics — Second Secondary

Definition and conditions on the base

An exponential function has the form f(x) = a^x, where a > 0 and a ≠ 1. The conditions have clear reasons:

If a = 1, then f(x) = 1 for every x. This is a constant function, not an exponential one.
If a < 0, then values such as (−4)^1/2 are not real numbers, so the base must be positive.
If a = 0, the expression is not defined for negative exponents.
The domain of an exponential function is the set of real numbers, and its range is ]0, ∞[. Its values are always positive.

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