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**Unformatted text preview: **h we have constructed are in order. As x → −∞
1 Take a class in Diﬀerential Equations and you’ll see why. 330 Exponential and Logarithmic Functions and attains values like x = −100 or x = −1000, the function f (x) = 2x takes on values like
1
1
f (−100) = 2−100 = 2100 or f (−1000) = 2−1000 = 21000 . In other words, as x → −∞,
2x ≈ 1
≈ very small (+)
very big (+) So as x → −∞, 2x → 0+ . This is represented graphically using the x-axis (the line y = 0) as a
horizontal asymptote. On the ﬂip side, as x → ∞, we ﬁnd f (100) = 2100 , f (1000) = 21000 , and so
on, thus 2x → ∞. As a result, our graph suggests the range of f is (0, ∞). The graph of f passes
the Horizontal Line Test which means f is one-to-one and hence invertible. We also note that when
we ‘connected the dots in a pleasing fashion’, we have made the implicit assumption that f (x) = 2x
is continuous2 and has a domain of all real numbers. In particular, we have suggested that things
√
√
like 2 3 exist as real numbers. We should take a moment to discuss what something like 2 3 might
mean, an...

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