Unformatted text preview: −2 is not a real number. In general, if x
is any rational number with an even denominator, then (−2)x is not deﬁned, so we must restrict
our attention to bases b ≥ 0. What about b = 0? The function f (x) = 0x is undeﬁned for x ≤ 0
because we cannot divide by 0 and 00 is an indeterminant form. For x > 0, 0x = 0 so the function
f (x) = 0x is the same as the function f (x) = 0, x > 0. We know everything we can possibly know
about this function, so we exclude it from our investigations. The only other base we exclude is
b = 1, since the function f (x) = 1x = 1 is, once again, a function we have already studied. We are
now ready for our deﬁnition of exponential functions.
Definition 6.1. A function of the form f (x) = bx where b is a ﬁxed real number, b > 0, b = 1 is
called a base b exponential function.
We leave it to the reader to verify6 that if b > 1, then the exponential function f (x) = bx will share
the same basic shape and characteristics as f (x) = 2x . What if 0 < b < 1? Consider g (x) = 1 .
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