mac1140_lecture22_1

mac1140_lecture22_1 - L22 5.2 Exponential Functions...

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191 L22 §5.2 Exponential Functions Definition . The exponential function with base a is a function of the form () x fx a = , where 0 a > and 1 a . Note: 0 (0) fa == ____ So, the point ______ is on the graph of ( ) fx . Reviewing some Rules : Example . Write each expression in the form x a : 3 (2 ) x = 2 (3 )(9 ) xx = 21 2 27 9 x x + =
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192 Some additional properties of exponents: Let 0 a > and 1 a , then: 1. For any real x , x a is a unique real number. Thu s , x ya = is a function. 2. bc aa b c =⇔ = s , x = is a 1-1 function. Note : Property 2 is helpful for solving exponential equations . Example : Solve the equations: a) 1 11 24 x x −+ ⎛⎞ = ⎜⎟ ⎝⎠ b) 1 1 27 9( 3 ) x x =
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193 Graphing the Exponential Function : Below are the graphs of: 2 x y = and 1 2 x y ⎛⎞ = ⎜⎟ ⎝⎠ . Use these as templates for basic graphs of x ya = : the graph on the left – for 1 a > ; the graph on the right – for 01 a < < . Properties of the graph of () x fx a = : 1. Points (0,1) and (1, ) a are on the graph. 2. If 1 a > , then f is increasing; if a << , then f is decreasing.
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mac1140_lecture22_1 - L22 5.2 Exponential Functions...

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