7.1 - CHAPTER 7 Some real numbers satisfy polynomial equations with integer coefficients 3 5 THE TRANSCENDENTAL FUNCTIONS satisfies the equation 5x

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CHAPTER 7 THE TRANSCENDENTAL FUNCTIONS Some real numbers satisfy polynomial equations with integer coefFcients: 3 5 satisFes the equation 5 x 3 = 0, 2 satisFes the equation x 2 2 = 0. Such numbers are called algebraic. There are, however, numbers that are not algebraic, among them π . Such numbers are called transcendental . Some functions f satisfy polynomial equations with polynomial coefFcients: f ( x ) = x π x + 2 satisFes the equation ( π x + 2) f ( x ) x = 0, f ( x ) = 2 x 3 x 2 satisFes the equation [ f ( x )] 2 + 6 x 2 f ( x ) + (9 x 4 4 x ) = 0. Such functions are called algebraic .There are, however, functions that are not algebraic; these are called transcendental . ±or example, the trigonometric functions are tran- scendental functions. In this chapter we introduce other transcendental functions: the logarithm function, the exponential function, and the inverse trigonometric functions. But Frst, a little more on functions in general. ± 7.1 ONE-TO-ONE FUNCTIONS; INVERSES One-to-One Functions A function can take on the same value at different points of its domain. Constant functions, for example, take on the same value at all points of their domains. The quadratic function f ( x ) = x 2 takes on the same value at c as it does at c ; so does the absolute-value function g ( x ) =| x | . The function f ( x ) = 1 + ( x 3)( x 5) takes on the same value at x = 5 as it does at x = 3: f (3) = 1, f (5) = 1. 371
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372 ± CHAPTER 7 THE TRANSCENDENTAL FUNCTIONS Functions for which this kind of repetition does not occur are called one-to-one functions . DEFINITION 7.1.1 A function f is said to be one-to-one if there are no two distinct numbers in the domain of f at which f takes on the same value: f ( x 1 ) = f ( x 2 ) implies x 1 = x 2 . Thus, if f is one-to-one and x 1 , x 2 are different points of the domain, then f ( x 1 ) ±= f ( x 2 ). The functions f ( x ) = x 3 and f ( x ) = x are both one-to-one. The cubing function is one-to-one because no two distinct numbers have the same cube. The square-root function is one-to-one because no two distinct nonnegative numbers have the same square root. x y x 2 f ( x 1 ) = f ( x 2 ) f is not one-to-one: x 1 Figure 7.1.1 x y is one-to-one: Figure 7.1.2 There is a simple geometric test, called the horizontal line test, which can be used to determine whether a function is one-to-one. Look at the graph of the function. If some horizontal line intersects the graph more than once, then the function is not one-to-one (Figure 7.1.1). If, on the other hand, no horizontal line intersects the graph more than once, then the function is one-to-one (Figure 7.1.2). Inverses We begin with a theorem about one-to-one functions. THEOREM 7.1.2 If f is a one-to-one function, then there is one and only one function g with domain equal to the range of f that satis±es the equation f ( g ( x )) = x for all x in the range of f .
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This note was uploaded on 03/28/2010 for the course MATH 1334 taught by Professor Mill during the Spring '10 term at Aarhus Universitet.

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7.1 - CHAPTER 7 Some real numbers satisfy polynomial equations with integer coefficients 3 5 THE TRANSCENDENTAL FUNCTIONS satisfies the equation 5x

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