4_2 - 1 and(1 a The number e is an IRRATIONAL number and...

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Section 4.2: Exponential Functions Instructor: Ms. Hoa Nguyen ([email protected]) 1 Exponential Functions An exponential function f is of the form f ( x ) = a x with a > 0, a 6 = 1. It is one-to-one . The domain of f is the set of ALL real numbers. The range is the set of positive real numbers . Questions: If a = 1, then what is f ( x ) = a x ? If a < 0, then what happens to f ( x ) = a x when x = 1 2 ? Remark : y = a x is an EXPONENTIAL function with the condition that a is a real number, a > 0, a 6 = 1. y = x n is a POWER function with the condition that n is a positive integer. Properties of exponentials ( a, b > 0 ) : a 0 = 1 a s + t = a s · a t a s · t = ( a s ) t ( ab ) s = a s b s a - s = 1 a s = ( 1 a ) s 1
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2 The graph of f ( x ) = a x , a > 1 The graph of f ( x ) = 2 x ( a > 1) The graph of y = a x , a > 1 is similar to the graph of y = 2 x . The graph is strictly increasing , i.e., if x 1 < x 2 then a x 1 < a x 2 . y → ∞ as x → ∞ y 0 as x → -∞ The x -axis ( y = 0) is a horizontal asymptote. The y -intercept is 1. The graph is continuous (no gaps) and smooth (no corners). The graph of f contains the points (0
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Unformatted text preview: , 1) and (1 , a ). The number e is an IRRATIONAL number, and approximately equal to 2 . 718. The graphing calculators have the key e x or exp( x ) to compute the exponential function y = a x where a is equal to the number e . 2 3 The graph of f ( x ) = a x , < a < 1 The graph of f ( x ) = ( 1 2 ) x (0 < a < 1) • The graph of y = a x , 0 < a < 1 is similar to the graph of y = ( 1 2 ) x . • The graph is strictly decreasing , i.e., if x 1 < x 2 then a x 1 > a x 2 . • y → 0 as x → ∞ • y → ∞ as x → -∞ • The x-axis ( y = 0) is a horizontal asymptote. • The y-intercept is 1. • The graph is continuous (no gaps) and smooth (no corners). • The graph of f contains the points (0 , 1) and (1 , a ). 3 4 Examples Question 1: Question 2: 4...
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4_2 - 1 and(1 a The number e is an IRRATIONAL number and...

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