FUNCTIONS AND THEIR GRAPHS in .NET

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FUNCTIONS AND THEIR GRAPHS
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Applying the quadratic formula (see the algebra insert in Problem 5.2), ( 2)2 4(10) 2 36 2 36 1 = = 2 2 2 2 6 1 = 1 3 1 = 2 Thus, the other two roots of f (x) are the complex numbers 1 + 3 1 and 1 3 1. x= ( 2) (b) Let us nd the roots of f (x) = x 3 5x 2 + 3x + 9. The integral roots (if any) must be divisors of 9: 1, 3, 9. 1 is not a root, but 1 is a root. Hence f (x) is divisible by x + 1 [see Fig. 7-10(b)]. The other roots of f (x) must be the roots of x 2 6x + 9. But x 2 6x + 9 = (x 3)2 . Thus, 1 and 3 are the roots of f (x); 3 is called a repeated root because (x 3)2 is a factor of x 3 5x 2 + 3x + 9.
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Theorem 7.3: (Fundamental Theorem of Algebra): of degree n has exactly n roots.
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If repeated roots are counted multiply, then every polynomial
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EXAMPLE In example (b) above, the polynomial x3 5x2 + 3x + 9 of degree 3 has two roots, 1 and 3, but 3 is a repeated
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root of multiplicity 2 and, therefore, is counted twice. A proof of Theorem 7.3 cannot be given here because it requires knowledge of the theory of functions of a complex variable.
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Since the complex roots of a polynomial with real coef cients occur in pairs, a b 1, the polynomial can have only an even number (possibly zero) of complex roots. Hence, a polynomial of odd degree must have at least one real root.
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7.1 Find the domain and the range of the function f such that f (x) = x 2 .
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Since x 2 is de ned for every real number x, the domain of f consists of all real numbers. To nd the range, notice that x 2 0 for all x and, therefore, x 2 0 for all x. Every nonpositive number y appears as a value x 2 for a suitable argument x; namely, for the argument x = y (and also for the argument x = y). Thus the range of f is ( , 0]. This can be seen more easily by looking at the graph of y = x 2 [see Fig. 7-11(a)].
Fig. 7-11 7.2 Find the domain and the range of the function f de ned by f (x) = x + 1 if 1 < x < 0 x if 0 x < 1
CHAP. 7]
FUNCTIONS AND THEIR GRAPHS
The domain of f consists of all x such that either 1 < x < 0 or 0 x < 1. This makes up the open interval ( 1, 1). The range of f is easily found from the graph in Fig. 7-11(b), whose projection onto the y-axis is the half-open interval [0, 1).
7.3 De ne f (x) as the greatest integer less than or equal to x; this value is usually denoted by [x]. Find the domain and the range, and draw the graph of f .
Since [x] is de ned for all x, the domain is the set of all real numbers. The range of f consists of all integers. Part of the graph is shown in Fig. 7-12. It consists of a sequence of horizontal, half-open unit intervals. (A function whose graph consists of horizontal segments is called a step function.)
Fig. 7-12
Fig. 7-13
7.4 Consider the function f de ned by the formula f (x) = x2 1 x 1
whenever this formula makes sense. Find the domain and the range, and draw the graph of f .
The formula makes sense whenever the denominator x 1 is not 0. Hence, the domain of f is the set of all real numbers different from 1. Now x 2 1 = (x 1)(x + 1) and so x2 1 =x+1 x 1 Hence, the graph of f (x) is the same as the graph of y = x + 1, except that the point corresponding to x = 1 is missing (see Fig. 7-13). Thus, the range consists of all real numbers except 2.
7.5 (a) Show that a set of points in the xy-plane is the graph of some function of x if and only if the set intersects every vertical line in at most one point (vertical line test). (b) Determine whether the sets of points indicated in Fig. 7-14 are graphs of functions.
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