1. Chapter 1 Class 12 Relation and Functions
2. Serial order wise
3. Ex 1.3

Transcript

Ex 1.3 , 6 Show that f: [โ1, 1] โ R, given by f(x) = ๐ฅ/(๐ฅ + 2) is one-one. Find the inverse of the function f: [โ1, 1] โ Range f. (Hint: For y โ Range f, y = f(x) = ๐ฅ/(๐ฅ + 2) , for some x in [โ1, 1], i.e., x = 2๐ฆ/(1 โ ๐ฆ) ) f(x) = x/(x+2) Check one-one f(x1) = ๐ฅ1/(๐ฅ1 + 2) f(x2) = ๐ฅ2/(๐ฅ2 + 2) Putting f(x1) = f(x2) ๐ฅ1/(๐ฅ1 + 2) = ๐ฅ2/(๐ฅ2 + 2) x1(x2 + 2) = x2(x1 + 2) x1x2 + 2x1 = x2x1 + 2x2 x1x2 โ x2x1 + 2x1 = 2x2 0 + 2x1 = 2x2 2x1 = 2x2 โ x1 = x2 Hence, if f(x1) = f(x2) , then x1 = x2 โด f is one-one Finding inverse f(x) = ๐ฅ/(๐ฅ + 2) Putting f(x) = y y = ๐ฅ/(๐ฅ + 2) y(x + 2) = x yx + 2y = x yx โ x = โ2y x(y โ 1) = โ2y x = (โ2๐ฆ )/(๐ฆ โ1) x = (โ2๐ฆ )/(โ1(โ๐ฆ + 1) ) x = (2๐ฆ )/((1 โ ๐ฆ) ) Let g(y) = (2๐ฆ )/((1 โ ๐ฆ) ) where g: R โ [โ1, 1], y โ  1 Thus, g is inverse of f Inverse of f = g(y) = (๐๐ )/((๐ โ ๐) ) , y โ  1

Ex 1.3