# What is the domain of f(x) = (-2)^x?

Jan 18, 2017

Explanation below.

#### Explanation:

The domain of any exponential function of the form ${a}^{x}$ is $\mathbb{R}$, provided $0 < a \ne 1$, unless specified otherwise. However, in our case, $- 2 < 0$, which creates some problems. Technically, the domain is still $\mathbb{R}$, since for any $x$, $f$ is defined, but we do not really know what happens for non-integer values of $x$ - will $- 2$ become positive with exponentiation, or will it be negative? On top of that, $f$ will alternate between positive and negative with every integer value its graph crosses.

There are varying opinions about those types of "semi-nonsensical" functions, so, to avoid those problems, it is quite often the case that the definition of exponential functions requires the base to be positive, and even different than one, so that the function will have a purpose.