Stitz-Zeager_College_Algebra_e-book

Next we see the quotient rule is applicable with u 3

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Unformatted text preview: certainly build a table of values and connect the points, or we could take a step back and x 1x note that g (x) = 2 = 2−1 = 2−x = f (−x), where f (x) = 2x . Thinking back to Section 1.8, 2 Recall that this means there are no holes or other kinds of breaks in the graph. You can actually prove this by considering the polynomial p(x) = x2 − 3 and showing it has no rational zeros by applying Theorem 3.9. 4 This is where Calculus and continuity come into play. 5 Want more information? Look up “convergent sequences” on the Internet. 6 Meaning, graph some more examples on your own. 3 6.1 Introduction to Exponential and Logarithmic Functions 331 the graph of f (−x) is obtained from the graph of f (x) by reflecting it across the y -axis. As such, we have y y 8 7 7 6 6 5 5 4 4 3 3 2 2 1 −3 −2 −1 8 1 1 2 3 reflect across y -axis x y = f (x) = 2x −3 −2 −1 −− − − − −→ −−−−−− multiply each x-coordinate by −1 1 2 3 y = g (x) = 2−x = x 1x 2 We see that the domain and range of g match that of f , namely (−∞, ∞) and (0, ∞), respectively. Like f , g is also one-to-one. Whereas f is always increasing, g is always decreasing. As a result, as x → −∞, g (x) → ∞, and on the flip side, as x → ∞, g (x) → 0+ . It shouldn’t be too surprising that for all choices of the base 0 <...
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