Ex 1.3, 6 - Show f(x) = x/x+2 is one-one. Find inverse of f. - Finding Inverse

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  1. Class 12
  2. Important Question for exams Class 12

Transcript

Ex 1.3 , 6 Show that f: [โˆ’1, 1] โ†’ R, given by f(x) = ๐‘ฅ/(๐‘ฅ + 2) is one-one. Find the inverse of the function f: [โˆ’1, 1] โ†’ Range f. (Hint: For y โˆˆ Range f, y = f(x) = ๐‘ฅ/(๐‘ฅ + 2) , for some x in [โˆ’1, 1], i.e., x = 2๐‘ฆ/(1 โˆ’ 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 โˆด 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 = (โˆ’2๐‘ฆ )/(๐‘ฆ โˆ’1) x = (โˆ’2๐‘ฆ )/(โˆ’1(โˆ’๐‘ฆ + 1) ) x = (2๐‘ฆ )/((1 โˆ’ ๐‘ฆ) ) Let g(y) = (2๐‘ฆ )/((1 โˆ’ ๐‘ฆ) ) where g: R โ†’ [โˆ’1, 1], y โ‰  1 Thus, g is inverse of f Inverse of f = g(y) = (๐Ÿ๐’š )/((๐Ÿ โˆ’ ๐’š) ) , y โ‰  1

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 9 years. He provides courses for Maths and Science at Teachoo.