Misc 2 - Let f(n) = n - 1, if is odd, f(n) = n + 1, if even - Finding Inverse

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

Transcript

Misc 2 (Method 1) Let f: W โ†’ W be defined as f(n) = n โˆ’ 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‘›โˆ’1 , ๐‘–๐‘“ ๐‘› ๐‘–๐‘  ๐‘œ๐‘‘๐‘‘๏ทฎ๐‘›+1, ๐‘–๐‘“ ๐‘› ๐‘–๐‘  ๐‘’๐‘ฃ๐‘’๐‘›๏ทฏ๏ทฏ Step 1 Let f(n) = y , such that y โˆˆ W n = ๐‘ฆโˆ’1, ๐‘–๐‘“ ๐‘ฆ ๐‘–๐‘  ๐‘œ๐‘‘๐‘‘๏ทฎ๐‘ฆ+1 , ๐‘–๐‘“ ๐‘ฆ ๐‘–๐‘  ๐‘’๐‘ฃ๐‘’๐‘›๏ทฏ๏ทฏ Let g(y) = ๐‘ฆโˆ’1, ๐‘–๐‘“ ๐‘ฆ ๐‘–๐‘  ๐‘œ๐‘‘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)) โˆด gof = n = IW Now, f(n) = ๐‘›โˆ’1 , ๐‘–๐‘“ ๐‘› ๐‘–๐‘  ๐‘œ๐‘‘๐‘‘๏ทฎ๐‘›+1, ๐‘–๐‘“ ๐‘› ๐‘–๐‘  ๐‘’๐‘ฃ๐‘’๐‘›๏ทฏ๏ทฏ & g(y) = ๐‘ฆโˆ’1, ๐‘–๐‘“ ๐‘ฆ ๐‘–๐‘  ๐‘œ๐‘‘๐‘‘๏ทฎ๐‘ฆ+1 , ๐‘–๐‘“ ๐‘ฆ ๐‘–๐‘  ๐‘’๐‘ฃ๐‘’๐‘›๏ทฏ๏ทฏ Step 3: fog = f(g(y)) โˆด fog = y = IW Since gof = IW and fog =IW f is invertible and inverse of f = g(y) = ๐‘ฆโˆ’1, ๐‘–๐‘“ ๐‘ฆ ๐‘–๐‘  ๐‘œ๐‘‘๐‘‘๏ทฎ๐‘ฆ+1 , ๐‘–๐‘“ ๐‘ฆ ๐‘–๐‘  ๐‘’๐‘ฃ๐‘’๐‘›๏ทฏ๏ทฏ Now g(y) = ๐‘ฆโˆ’1, ๐‘–๐‘“ ๐‘ฆ ๐‘–๐‘  ๐‘œ๐‘‘๐‘‘๏ทฎ๐‘ฆ+1 , ๐‘–๐‘“ ๐‘ฆ ๐‘–๐‘  ๐‘’๐‘ฃ๐‘’๐‘›๏ทฏ๏ทฏ Replacing y with n g(n) = ๐‘›โˆ’1 , ๐‘–๐‘“ ๐‘› ๐‘–๐‘  ๐‘œ๐‘‘๐‘‘๏ทฎ๐‘›+1, ๐‘–๐‘“ ๐‘› ๐‘–๐‘  ๐‘’๐‘ฃ๐‘’๐‘›๏ทฏ๏ทฏ = f(n) โˆด Inverse of f is f itself Misc 2 (Method 2) Let f: W โ†’ W be defined as f(n) = n โˆ’ 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‘›โˆ’1 , ๐‘–๐‘“ ๐‘› ๐‘–๐‘  ๐‘œ๐‘‘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 โˆด function f is one-one Check onto f(n) = ๐‘›โˆ’1 , ๐‘–๐‘“ ๐‘› ๐‘–๐‘  ๐‘œ๐‘‘๐‘‘๏ทฎ๐‘›+1, ๐‘–๐‘“ ๐‘› ๐‘–๐‘  ๐‘’๐‘ฃ๐‘’๐‘›๏ทฏ๏ทฏ Let f(n) = y , such that y โˆˆ W n = ๐‘ฆโˆ’1, ๐‘–๐‘“ ๐‘ฆ ๐‘–๐‘  ๐‘œ๐‘‘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 โˆˆ W Thus, f is onto. Finding inverse f(n) = ๐‘›โˆ’1 , ๐‘–๐‘“ ๐‘› ๐‘–๐‘  ๐‘œ๐‘‘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‘ฆโˆ’1, ๐‘–๐‘“ ๐‘ฆ ๐‘–๐‘  ๐‘œ๐‘‘๐‘‘๏ทฎ๐‘ฆ+1 , ๐‘–๐‘“ ๐‘ฆ ๐‘–๐‘  ๐‘’๐‘ฃ๐‘’๐‘›๏ทฏ๏ทฏ โˆด Inverse of f = g(y) = ๐‘ฆโˆ’1, ๐‘–๐‘“ ๐‘ฆ ๐‘–๐‘  ๐‘œ๐‘‘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‘ฆโˆ’1, ๐‘–๐‘“ ๐‘ฆ ๐‘–๐‘  ๐‘œ๐‘‘๐‘‘๏ทฎ๐‘ฆ+1 , ๐‘–๐‘“ ๐‘ฆ ๐‘–๐‘  ๐‘’๐‘ฃ๐‘’๐‘›๏ทฏ๏ทฏ Replacing y with n g(n) = ๐‘›โˆ’1 , ๐‘–๐‘“ ๐‘› ๐‘–๐‘  ๐‘œ๐‘‘๐‘‘๏ทฎ๐‘›+1, ๐‘–๐‘“ ๐‘› ๐‘–๐‘  ๐‘’๐‘ฃ๐‘’๐‘›๏ทฏ๏ทฏ = f(n) โˆด Inverse of f is f itself

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 8 years. He provides courses for Maths and Science at Teachoo. You can check his NCERT Solutions from Class 6 to 12.