rogawskilt4e_lectureslides_ch07.2.pptx - Chapter 7...

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Chapter 7: EXPONENTIAL AND LOGARITHMIC FUNCTIONS Contents: 7.1 The Derivative of f ( x ) = b x and the Number e 7.2 Inverse Functions 7.3 Logarithmic Functions and Their Derivatives 7.4 Applications of Exponential and Logarithmic Functions 7.5 L’Hôpital’s Rule 7.6 Inverse Trigonometric Functions 7.7 Hyperbolic Functions
The inverse of a function f , denoted f 1 , is the function that reverses the effect of f (Figure 1). For example, the inverse of f ( x ) = x 3 is the cube root function f −1 ( x ) = x 1/3 . DEFINITION : Inverse Let f have domain D and range R . If there is a function g with domain R such that g ( f ( x )) = x for all x D and f ( g ( x )) = x for all x R then f is said to be invertible . The function g is called the inverse function and is denoted f −1 . 7.2 Inverse Functions (1 of 14)
EXAMPLE : We show that f ( x ) = 2 x − 18 is invertible by computing the inverse function in two steps. Step 1. Solve the equation y = f ( x ) for x in terms of y . y = 2 x − 18 y + 18 = 2 x x = y + 9 This gives us the inverse as a function of the variable y : f −1 ( y ) = y + 9. Step 2. Interchange variables. We usually prefer to write the inverse as a function of x , so we interchange the roles of x and y : f −1 ( x ) = x + 9 7.2 Inverse Functions (2 of 14)
Graphs of f and f −1 are shown in Figure 2. To check our calculation, let’s verify that f −1 ( f ( x )) = x and f ( f −1 ( x )) = x : Because f −1 is a linear function, its domain and range are R . 7.2 Inverse Functions (3 of 14)
Let us now look at f ( x ) = x 2 , and let us attempt to find an inverse function f − 1 ( x ) . The problem is that

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