Finding Inverse

Chapter 1 Class 12 Relation and Functions
Concept wise

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Question 2(Method 1) Let f: W โ W be defined as f(n) = n โ 1, if is odd and f(n) = n + 1, if n is even. Show that f is invertible. Find the inverse of f. Here, W is the set of all whole numbers. f(n) = ๐โ1 , ๐๐ ๐ ๐๐  ๐๐๐๏ทฎ๐+1, ๐๐ ๐ ๐๐  ๐๐ฃ๐๐๏ทฏ๏ทฏ Step 1 Let f(n) = y , such that y โ W n = ๐ฆโ1, ๐๐ ๐ฆ ๐๐  ๐๐๐๏ทฎ๐ฆ+1 , ๐๐ ๐ฆ ๐๐  ๐๐ฃ๐๐๏ทฏ๏ทฏ Let g(y) = ๐ฆโ1, ๐๐ ๐ฆ ๐๐  ๐๐๐๏ทฎ๐ฆ+1 , ๐๐ ๐ฆ ๐๐  ๐๐ฃ๐๐๏ทฏ๏ทฏ where g: W โ W Step 2: gof = g(f(n)) โด gof = n = IW Now, f(n) = ๐โ1 , ๐๐ ๐ ๐๐  ๐๐๐๏ทฎ๐+1, ๐๐ ๐ ๐๐  ๐๐ฃ๐๐๏ทฏ๏ทฏ & g(y) = ๐ฆโ1, ๐๐ ๐ฆ ๐๐  ๐๐๐๏ทฎ๐ฆ+1 , ๐๐ ๐ฆ ๐๐  ๐๐ฃ๐๐๏ทฏ๏ทฏ Step 3: fog = f(g(y)) โด fog = y = IW Since gof = IW and fog =IW f is invertible and inverse of f = g(y) = ๐ฆโ1, ๐๐ ๐ฆ ๐๐  ๐๐๐๏ทฎ๐ฆ+1 , ๐๐ ๐ฆ ๐๐  ๐๐ฃ๐๐๏ทฏ๏ทฏ Now g(y) = ๐ฆโ1, ๐๐ ๐ฆ ๐๐  ๐๐๐๏ทฎ๐ฆ+1 , ๐๐ ๐ฆ ๐๐  ๐๐ฃ๐๐๏ทฏ๏ทฏ Replacing y with n g(n) = ๐โ1 , ๐๐ ๐ ๐๐  ๐๐๐๏ทฎ๐+1, ๐๐ ๐ ๐๐  ๐๐ฃ๐๐๏ทฏ๏ทฏ = f(n) โด Inverse of f is f itself Misc 2 (Method 2) Let f: W โ W be defined as f(n) = n โ 1, if is odd and f(n) = n + 1, if n is even. Show that f is invertible. Find the inverse of f. Here, W is the set of all whole numbers. f(n) = ๐โ1 , ๐๐ ๐ ๐๐  ๐๐๐๏ทฎ๐+1, ๐๐ ๐ ๐๐  ๐๐ฃ๐๐๏ทฏ๏ทฏ f is invertible if f is one-one and onto Check one-one There can be 3 cases โข x1 & x2 both are odd โข x1 & x2 both are even โข x1 is odd & x2 is even If x1 & x2 are both odd f(x1) = x1 + 1 f(x2) = x2 + 1 Putting f(x1) = f(x2) x1 + 1 = x2 + 1 x1 = x2 If x1 & x2 are both are even f(x1) = x1 โ 1 f(x2) = x2 โ 1 If f(x1) = f(x2) x1 โ 1 = x2 โ 1 x1 = x2 If x1 is odd and x2 is even f(x1) = x1 + 1 f(x2) = x2 โ 1 If f(x1) = f(x2) x1 + 1 = x2 โ 1 x2 โ x1 = 2 which is impossible as difference between even and odd number can never be even Hence, if f(x1) = f(x2) , x1 = x2 โด function f is one-one Check onto f(n) = ๐โ1 , ๐๐ ๐ ๐๐  ๐๐๐๏ทฎ๐+1, ๐๐ ๐ ๐๐  ๐๐ฃ๐๐๏ทฏ๏ทฏ Let f(n) = y , such that y โ W n = ๐ฆโ1, ๐๐ ๐ฆ ๐๐  ๐๐๐๏ทฎ๐ฆ+1 , ๐๐ ๐ฆ ๐๐  ๐๐ฃ๐๐๏ทฏ๏ทฏ Hence, if y is a whole number, n will also be a whole number i.e. n โ W Thus, f is onto. Finding inverse f(n) = ๐โ1 , ๐๐ ๐ ๐๐  ๐๐๐๏ทฎ๐+1, ๐๐ ๐ ๐๐  ๐๐ฃ๐๐๏ทฏ๏ทฏ For finding inverse, we put f(n) = y and find n in terms of y We have done that while proving onto n = ๐ฆโ1, ๐๐ ๐ฆ ๐๐  ๐๐๐๏ทฎ๐ฆ+1 , ๐๐ ๐ฆ ๐๐  ๐๐ฃ๐๐๏ทฏ๏ทฏ โด Inverse of f = g(y) = ๐ฆโ1, ๐๐ ๐ฆ ๐๐  ๐๐๐๏ทฎ๐ฆ+1 , ๐๐ ๐ฆ ๐๐  ๐๐ฃ๐๐๏ทฏ๏ทฏ where g: W โ W Now g(y) = ๐ฆโ1, ๐๐ ๐ฆ ๐๐  ๐๐๐๏ทฎ๐ฆ+1 , ๐๐ ๐ฆ ๐๐  ๐๐ฃ๐๐๏ทฏ๏ทฏ Replacing y with n g(n) = ๐โ1 , ๐๐ ๐ ๐๐  ๐๐๐๏ทฎ๐+1, ๐๐ ๐ ๐๐  ๐๐ฃ๐๐๏ทฏ๏ทฏ = f(n) โด Inverse of f is f itself