516 CHAPTER 4 Inverse, Exponential, and Logarithmic Functions Chapter 4Test Prep Key Terms 4.1 one-to-one function inverse function 4.2 exponential function exponential equation compound interest future value present value compound amount continuous compounding 4.3 logarithm base argument logarithmic equation logarithmic function 4.4 common logarithm pH natural logarithm 4.6 doubling time half-life New Symbols ƒ−11x2 inverse of ƒ1x2 e a constant, approximately 2.718281828459045 log a x logarithm of x with base a log x common (base 10) logarithm of x ln x natural (base e) logarithm of x Quick Review Concepts Examples The function y = ƒ1x2 = x2 is not one-to-one, because y = 16, for example, corresponds to both x = 4 and x = -4. The graph of ƒ1x2 = 2x - 1 is a straight line with slope 2. ƒ is a one-to-one function by the horizontal line test. Find the inverse of ƒ. ƒ1x2 = 2x - 1 Given function y = 2x - 1 Let y = ƒ1x2. x = 2y - 1 Interchange x and y. y = x + 1 2 Solve for y. ƒ-11x2 = x + 1 2 Replace y with ƒ-11x2. ƒ-11x2 = 1 2 x + 1 2 x + 1 2 = x 2 + 1 2 = 1 2 x + 1 2 4.1 Inverse Functions One-to-One Function In a one-to-one function, each x-value corresponds to only one y-value, and each y-value corresponds to only one x-value. A function ƒ is one-to-one if, for elements a and b in the domain of ƒ, a 3b implies ƒ1a2 3ƒ1b2. Horizontal Line Test A function is one-to-one if every horizontal line intersects the graph of the function at most once. Inverse Functions Let ƒ be a one-to-one function. Then g is the inverse function of ƒ if 1 ƒ° g2 1x2 =x for every x in the domain of g and 1g° ƒ2 1x2 =x for every x in the domain of ƒ. Equation of the Inverse of y =ƒ1x2 For a one-to-one function y = ƒ1x2, the equation of the inverse is found as follows. Step 1 Interchange x and y. Step 2 Solve for y. Step 3 Replace y with ƒ-11x2.
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