# qqm04.pdf - Managerial Mathematics SQQM1023 4.1...

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Managerial Mathematics: : SQQM1023 1 4.1: INTRODUCTION TO EXPONENTIAL FUNCTION An exponential function involves a constant (base) raised to a variable power (exponent) x , t and so on. DEFINITION 4.1.1(a): Exponential function with base a The function f is defined by   x a x f y where 0 a , 1 a and the exsponent x is any real number is called an exponential function with base . a TAKRIF 4.1.1(b): Exponential function with base e (2.71828...) The function f is defined by   x e x f y where ... 71828 . 2 e , and the exponent x is any real numbers is called an exponential function with base . e or natural exponential function . Determine whether the given functions are an exponential function or not. If it is, then determine the base for each function. a) x y 2 b)   t t g 3 c) t t y 3 d) 3 / 11 x y e) x e y 05 . 0 f)   x x h 2 . 1 5 . 0 g) x x g 6 718 . 2 ) ( Example 1

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Managerial Mathematics: : SQQM1023 2 4.1.2 SKETCHING AN EXPONENTIAL GRAPH. Let say x a y and 1 a for x . Therefore the graph for x a y can be sketch by replacing several values for x . a) If 2 a . Then the exponential function would be x y 2 Build a table with several values of x, and find the corresponding values for y: x -2 -1 0 1 2 x y 2 Based on the table, we got several points that resides on the graph. Therefore, we just plot all the points and draw a curve that connects all those points. 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 -3 -2 -1 0 1 2 3
Managerial Mathematics: : SQQM1023 3 b) Now let say 2 1 a . The exponential function would be x y 2 1 Try to sketch the graph for this function. i. Build a table: x -2 -1 0 1 2 1 2 x y ii. Based on the table, plot all of the points and draw a curve: c) If the exponential function has the base . e Where x y , therefore Build a table: x -2 -1 0 1 2 x e y Therefore the graph would be: 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 -3 -2 -1 0 1 2 3

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Managerial Mathematics: : SQQM1023 4 d) For the exponential function x y x -2 -1 0 1 2 x e y And the graph would be: 0 1 2 3 4 5 6 7 8 -3 -2 -1 0 1 2 3 0 1 2 3 4 5 6 7 8 -3 -2 -1 0 1 2 3
Managerial Mathematics: : SQQM1023 5 4.1.3 PROPERTIES OF EXPONENTIAL FUNCTIONS If a and b is any positive real numbers   and y x , is any rational number, therefore Property 1: y x y x a a a Property 2 : y x y x a a a Simplify Simplify a) 3 2x .
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• Spring '16
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