A Treatise on Differential Equations by George Boole

By George Boole

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Definition: For any functionf, a zero or root offis a number r such thatf(r) = 0. 52 FUNCTIONS AND THEIR GRAPHS [CHAP. ,a,, a, are integers), and if r is an integer that is a root off, then r must be a divisor of the constant term a,. 1, any integral root of x3 - 2x2 - 5x 6 must be among the divisors of 6, which are f1, f 2, f 3, and f 6. By actual substitution, it is found that 1, -2, and 3 are roots. 2: A number r is a root of the polynomial f ( x ) = a,x" + a , - , x n - ~+ * * * + a,x + a0 if and only iff(x) is divisible by the polynomial x - r.

7-6 is (0, 7-5(b)is ( - CO, 13. CO). The domain of the function graphed in Fig. 3 EVEN A N D ODD FUNCTIONS A function f is called even if, for any x in the domain off, - x is also in the domain off and f(-4 = f ( x ) . EXAMPLES (a) Letf(x) = x2 for all x. Then f(-x) = (-Xy = x2 =f(x) and sof is even. (b) Letf(x) = 3x4 - 5x2 + 2 for all x. Then f(-x) = 3(-x)4 - 5 ( - X y + 2 = 3x4 - 5x2 + 2 = f ( x ) Thus, f is even. More generally, a function that is defined for all x and involves only even powers of x is an even function.

7-6, in which projection of the graph onto the y-axis produces all y such that y > 0. I 1 1 ) X Fig. 7-6 Note: In many treatments of the foundations of mathematics, a function is identified with its graph. In other words, a function is defined to be a setfof ordered pairs such thatfdoes not contain two pairs (a, b) and (a, c) with b # c. Then “y = j ( x ) ) ’ is defined to mean the same thing as “(x, y) belongs to 5’’ However, this approach obscures the intuitive idea of a function. 2 INTERVALS In dealing with the domains and ranges of functions, intervals of numbers occur so often that it is convenient to introduce special notation and terminology for them.

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