Absolute Value of Modulus Function:

May 19th, 2015
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Domain R Range (0, ) Properties of modulus function Signum Function: (8) Greatest integer function: Properties of greatest integer function: (9) Fractional part function:

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Functions And GraphsPart-2Absolute Value of Modulus Function: y = |x| = Domain R Range (0, )Properties of modulus function: (i) |x| a - a x a (a 0)(ii) |x| a x - a or x a (a ) (iii) |x + y| |x| + |y|(iv) |x + y| Illustration: Find domain of y = Ans: y is defind if (|x| - x) > 0 |x| > x which hold for -ve x only Hence domain (- , 0)(7) Signum Function: y = Sgn(x) = Domain x R Range {- 1, 0, 1} (8) Greatest integer function: [x] indicates integral part of x which is nearest & smaller integer to x. It is also c/d floor of x or stepwise function.[2.3] = 2 [5] = 5, [- 0.6] = - 1 In general n n < n + 1 (n Integer) [n] = n x [x]0 x < 1 1 x < 2 2 x < 3 - 1 x < 0 - 2 x < - 1 0 1 2 - 1 - 2 Properties of greatest integer function: (i) [x] = x holds if x I I Integer. (ii) [x + I] = [x] + I if I is integer. (iii) [x + y] [x] + [y] (iv) If [(x)] I, then (x) I. (v) If [(x)] I, then (x) < I + 1 (vi) If [x] > n x n + 1, n I. (vii) [- x] = - [x] if x I & [- x] = - [x] - 1 if x I. (viii) [x + y] = [x] + [x + y - [x]] for all x, y R. (ix) [x] + = [n x], n N N natural no.Illustration: y = 2[x] + 3 & y = 3[x - 2] + 5 then find value of [x + y]. Ans: 2[x] + 3 = 3[x

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