FUNCTIONS AND THEIR GRAPHS in Visual Studio .NET

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FUNCTIONS AND THEIR GRAPHS
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Fig. 7-14
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(a) If a set of points is the graph of a function f , the set consists of all points (x, y), such that y = f (x). If (x0 , u) and (x0 , v) are points of intersection of the graph with the vertical line x = x0 , then u = f (x0 ) and v = f (x0 ). Hence, v = u, and the points are identical. Conversely, if a set A of points intersects each vertical line in at most one point, de ne a function f as follows. If A intersects the line x = x0 at some point (x0 , w), let f (x0 ) = w. Then A is the graph of f . (b) (i) and (iv) are not graphs of functions, since they intersect certain vertical lines in more than one point.
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7.6 Let f (x) = x 2 + 2x 1 for all x. Evaluate: (a) f (2) (e) f (x 1) (b) f ( 2) (f ) f (x + h) (c) f ( x) (g) f (x + h) f (x) (d) f (x + 1) f (x + h) f (x) (h) h
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In each case, we substitute the speci ed argument for all occurrences of x in the formula for f (x). (a) f (2) = (2)2 + 2(2) 1 = 4 + 4 1 = 7 (b) f ( 2) = ( 2)2 + 2( 2) 1 = 4 4 1 = 1 (c) f ( x) = ( x)2 + 2( x) 1 = x 2 2x 1 (d) f (x + 1) = (x + 1)2 + 2(x + 1) 1 = (x 2 + 2x + 1) + 2x + 2 1 = x 2 + 4x + 2 (e) f (x 1) = (x 1)2 + 2(x 1) 1 = (x 2 2x + 1) + 2x 2 1 = x 2 2 (f) f (x + h) = (x + h)2 + 2(x + h) 1 = (x 2 + 2hx + h2 ) + 2x + 2h 1 = x 2 + 2hx + 2x + h2 + 2h 1
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FUNCTIONS AND THEIR GRAPHS
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(g) Using the result of part ( f ), f (x + h) f (x) = (x 2 + 2hx + 2x + h2 + 2h 1) (x 2 + 2x 1) = x 2 + 2hx + 2x + h2 + 2h 1 x 2 2x + 1 = 2hx + h2 + 2h (h) Using the result of part (g), 2hx + h2 + 2h h(2x + h + 2) f (x + h) f (x) = = = 2x + h + 2 h h h
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7.7 Find all roots of the polynomial f (x) = x 3 8x 2 + 21x 20.
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The integral roots (if any) must be divisors of 20: 1, 2, 4, 5, 10, 20. Calculation shows that 4 is a root. Hence, x 4 must be a factor of f (x) (see Fig. 7-15). To nd the roots of x 2 4x + 5, use the quadratic formula 4 4 4 4 1 4 2 1 ( 4)2 4(5) = = = = 2 1 x= 2 2 2 2 Hence, there are one real root, 4, and two complex roots, 2 + 1 and 2 1. ( 4)
Fig. 7-15
Supplementary Problems
7.8 Find the domain and the range, and draw the graphs of the functions determined by the following formulas (for all arguments x for which the formulas make sense): (a) h(x) = 4 x 2 (b) G(x) = 2 x (c) H(x) = 4 x 2 J(x) = 4 x 2 (d) U(x) = x2 4 x (g) g(x) = 3 (e) V (x) = |x 1| 1 (h) h(x) = x (k) J(x) = (n) G(x) = x2 x 2 x2 4 x+2 if x 0 if x > 0 (f ) f (x) = [2x] 1 (i) F(x) = x 1 (l) U(x) = (o) H(x) = 3 x for x 1 5x 3 for x > 1 x 4 if x 2 if x > 2
1 ( j) H(x) = x 3 2 (m) f (1) = 1 f (2) = 3 f (4) = 1
FUNCTIONS AND THEIR GRAPHS 1 x if x 1 (q) g(x) = 2 if 1 < x < 1 2 x + 1 if x 1 (t) f (x) = 3 x 2 x 9 (r) h(x) = x 3 6
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