296 CHAPTER 5 Exponential and Logarithmic Functions Let’s find an equation ( ) = y f x that describes this function f. The key fact is that the value of f doubles for every 1-unit increase in x. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) = = = ⋅ = ⋅ = = ⋅ = ⋅ = = ⋅ = ⋅ = = ⋅ = ⋅ f f f f f f f f f 0 5 1 2 0 2 5 5 2 2 2 1 2 5 2 5 2 3 2 2 2 5 2 5 2 4 2 3 2 5 2 5 2 1 2 2 3 3 4 The pattern leads to ( ) ( ) ( ) = − = ⋅ = ⋅ − f x f x 2 1 2 5 2 5 2 x x 1 Double the value of f at 0 to get the value at 1. Double the value of f at 1 to get the value at 2. CAUTION It is important to distinguish a power function, ( ) = ≥ g x ax n , 2 n an integer, from an exponential function, ( ) = ⋅ ≠ > f x C a a a , 1, 0. x In a power function, the base is a variable and the exponent is a constant, as in = y x .2 In an exponential function, the base is a constant and the exponent is a variable, as in = y 2 .x j In the definition of an exponential function, the base = a 1 is excluded because this function is simply the constant function ( ) = ⋅ = f x C C 1 . x Bases that are negative are also excluded; otherwise, many values of x would have to be excluded from the domain, such as = x 1 2 and = x 3 4 . [Recall that ( ) ( ) ( ) − = − − = − = − 2 2, 3 3 27, 1 2 3 4 3 4 4 and so on, are not defined in the set of real numbers.] Transformations (vertical shifts, horizontal shifts, reflections, and so on) of a function of the form ( ) = f x Cax are also exponential functions. Examples of such exponential functions are ( ) ( ) ( ) ( ) = = + = ⋅ − f x F x G x 2 1 3 5 2 3 x x x 3 For each function, note that the base of the exponential expression is a constant and the exponent contains a variable. In the function ( ) = ⋅ f x 5 2 ,x notice that the ratio of consecutive outputs is constant for 1-unit increases in the input. This ratio equals the constant 2, the base of the exponential function. In other words, ( ) ( ) ( ) ( ) ( ) ( ) = ⋅ = = ⋅ ⋅ = = ⋅ ⋅ = f f f f f f 1 0 5 2 5 2 2 1 5 2 5 2 2 3 2 5 2 5 2 2 and so on 1 2 1 3 2 This leads to the following result. THEOREM For an exponential function ( ) = > ≠ ≠ f x Ca a a C , 0, 1, and 0, x if x is any real number, then ( ) ( ) ( ) ( ) + = + = f x f x a f x af x 1 or 1 DEFINITION Exponential Function An exponential function is a function of the form ( ) = f x Cax where a is a positive real number ( ) > ≠ a a 0 , 1, and ≠ C 0 is a real number. The domain of f is the set of all real numbers. The base a is the growth factor , and, because ( ) = = f Ca C C 0 , 0 is called the initial value .
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