Question

What happens when the base of an exponential function is negative?

Answer 1

in ℤ, successive values alternate signs

Explanation:

You want to know what happens when the base of an exponential function is negative.

Powers of a negative number

Consider the exponential function ...

y = a·b^x

For b < 0 and integer values of x, the signs of the corresponding values of y will alternate.

Example

y = 2·(-1)^x

A table of values will be ...

`\begin{array}{ccccccc}x&-1&0&1&2&3\\y&-2&2&-2&2&-2\end{array}`

Roots of a negative number

For non-integer values of x, the values of y become more complicated. For fractional values of x, where x has an even denominator, the equation does not produce a real value. Instead, you get a complex number.

For fractional values of x where x has an odd denominator, the power is defined, and the sign will depend on whether the numerator is even or odd.

For irrational values of x, the function value in general is complex.

Of course, there are an infinite number of rational numbers between any pair of numbers on the number line, so the graph will have both positive and negative values, but will not be continuous. The attached graph can give you the idea. It is not in any way a complete graph of the function, but plots selected values.

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Additional comment

The second attachment shows one of the uses of an exponential factor with a negative base. It creates the alternate signs of the series expansion of the sine function. Only whole number exponents are used in this case.