Stitz-Zeager_College_Algebra_e-book

# At rst it seems as if we have no means of simplifying

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Unformatted text preview: b < 1, the graph of y = bx behaves similarly to the graph of g . We summarize these observations, and more, in the following theorem whose proof ultimately requires Calculus. Theorem 6.1. Properties of Exponential Functions: Suppose f (x) = bx . • The domain of f is (−∞, ∞) and the range of f is (0, ∞). • (0, 1) is on the graph of f and y = 0 is a horizontal asymptote to the graph of f . • f is one-to-one, continuous and smootha • If 0 < b < 1: • If b > 1: – f is always increasing – f is always decreasing – As x → −∞, f (x) → 0+ – As x → −∞, f (x) → ∞ – As x → ∞, f (x) → ∞ – As x → ∞, f (x) → 0+ – The graph of f resembles: – The graph of f resembles: y = bx , b > 1 a y = bx , 0 < b < 1 Recall that this means the graph of f has no sharp turns or corners. 332 Exponential and Logarithmic Functions Of all of the bases for exponential functions, two occur the most often in scientiﬁc circles. The ﬁrst, base 10, is often called the common base. T...
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