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# Lect5 - Exponential Functions Definition An Exponential...

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92.131 Lecture 5 1 of 19 Ronald Brent © 2009 All rights reserved. −4 −3 −2 −1 1 2 3 4 0 1 2 3 4 5 6 7 8 9 10 y = 2 x y x Exponential Functions Definition: An Exponential Function is an function that has the form x a x f = ) ( , where a > 0. The number a is called the base . Example:Let x x f 2 ) ( = It is clear what the function means for some values of x . For example 1 2 ) 0 ( 0 = = f , 2 2 ) 1 ( 1 = = f , 4 2 ) 2 ( 2 = = f , 8 2 ) 3 ( 3 = = f , 2 1 2 ) 1 ( 1 = = f , 4 1 2 ) 2 ( 2 = = f , 414 . 1 2 2 ) ( 2 1 2 1 = = f , and 5 3 2 . 3 2 8 2 2 2 ) 2 . 3 ( 5 1 = = = f . Defining x x f 2 ) ( = for x irrational is too difficult now. This graph represents exponential growth since it is increasing as x increases.

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92.131 Lecture 5 2 of 19 Ronald Brent © 2009 All rights reserved. y −4 −3 −2 −1 1 2 3 4 0 1 2 3 4 5 6 7 8 9 10 y = (1/2) x = 2 - x y = 2 x x Recall that replacing x with – x in a function amounts to a rotation of the graph about the y -axis. So we can get the graph of x x x f = = 2 2 1 ) ( , from x x f 2 ) ( = shown previously. It is shown dotted below. The function x x f = 2 1 ) ( represents exponential decay since the function decreases as x increases.
92.131 Lecture 5 3 of 19 Ronald Brent © 2009 All rights reserved. x −5 −4 −3 −2 −1 0 1 2 3 4 5 y 1 2 3 4 5 6 7 8 9 10 Exponential Growth: All functions x a x f = ) ( where a > 1, exhibit exponential growth. x y 3 = x y 10 = x y 2 = x y 5 . 1 = x y 1 . 1 = x y 4 =

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