# Misc 2 - Chapter 1 Class 12 Relation and Functions (Term 1)

Last updated at May 29, 2018 by Teachoo

Last updated at May 29, 2018 by Teachoo

Transcript

Misc 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) = ๐โ1 , ๐๐ ๐ ๐๐ ๐๐๐๏ทฎ๐+1, ๐๐ ๐ ๐๐ ๐๐ฃ๐๐๏ทฏ๏ทฏ Step 1 Let f(n) = y , such that y โ W n = ๐ฆโ1, ๐๐ ๐ฆ ๐๐ ๐๐๐๏ทฎ๐ฆ+1 , ๐๐ ๐ฆ ๐๐ ๐๐ฃ๐๐๏ทฏ๏ทฏ Let g(y) = ๐ฆโ1, ๐๐ ๐ฆ ๐๐ ๐๐๐๏ทฎ๐ฆ+1 , ๐๐ ๐ฆ ๐๐ ๐๐ฃ๐๐๏ทฏ๏ทฏ where g: W โ W Step 2: gof = g(f(n)) โด gof = n = IW Now, f(n) = ๐โ1 , ๐๐ ๐ ๐๐ ๐๐๐๏ทฎ๐+1, ๐๐ ๐ ๐๐ ๐๐ฃ๐๐๏ทฏ๏ทฏ & g(y) = ๐ฆโ1, ๐๐ ๐ฆ ๐๐ ๐๐๐๏ทฎ๐ฆ+1 , ๐๐ ๐ฆ ๐๐ ๐๐ฃ๐๐๏ทฏ๏ทฏ Step 3: fog = f(g(y)) โด fog = y = IW Since gof = IW and fog =IW f is invertible and inverse of f = g(y) = ๐ฆโ1, ๐๐ ๐ฆ ๐๐ ๐๐๐๏ทฎ๐ฆ+1 , ๐๐ ๐ฆ ๐๐ ๐๐ฃ๐๐๏ทฏ๏ทฏ Now g(y) = ๐ฆโ1, ๐๐ ๐ฆ ๐๐ ๐๐๐๏ทฎ๐ฆ+1 , ๐๐ ๐ฆ ๐๐ ๐๐ฃ๐๐๏ทฏ๏ทฏ Replacing y with n g(n) = ๐โ1 , ๐๐ ๐ ๐๐ ๐๐๐๏ทฎ๐+1, ๐๐ ๐ ๐๐ ๐๐ฃ๐๐๏ทฏ๏ทฏ = f(n) โด 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) = ๐โ1 , ๐๐ ๐ ๐๐ ๐๐๐๏ทฎ๐+1, ๐๐ ๐ ๐๐ ๐๐ฃ๐๐๏ทฏ๏ทฏ f is invertible if f is one-one and onto Check one-one There can be 3 cases โข x1 & x2 both are odd โข x1 & x2 both are even โข 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 โด function f is one-one Check onto f(n) = ๐โ1 , ๐๐ ๐ ๐๐ ๐๐๐๏ทฎ๐+1, ๐๐ ๐ ๐๐ ๐๐ฃ๐๐๏ทฏ๏ทฏ Let f(n) = y , such that y โ W n = ๐ฆโ1, ๐๐ ๐ฆ ๐๐ ๐๐๐๏ทฎ๐ฆ+1 , ๐๐ ๐ฆ ๐๐ ๐๐ฃ๐๐๏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) = ๐โ1 , ๐๐ ๐ ๐๐ ๐๐๐๏ทฎ๐+1, ๐๐ ๐ ๐๐ ๐๐ฃ๐๐๏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 = ๐ฆโ1, ๐๐ ๐ฆ ๐๐ ๐๐๐๏ทฎ๐ฆ+1 , ๐๐ ๐ฆ ๐๐ ๐๐ฃ๐๐๏ทฏ๏ทฏ โด Inverse of f = g(y) = ๐ฆโ1, ๐๐ ๐ฆ ๐๐ ๐๐๐๏ทฎ๐ฆ+1 , ๐๐ ๐ฆ ๐๐ ๐๐ฃ๐๐๏ทฏ๏ทฏ where g: W โ W Now g(y) = ๐ฆโ1, ๐๐ ๐ฆ ๐๐ ๐๐๐๏ทฎ๐ฆ+1 , ๐๐ ๐ฆ ๐๐ ๐๐ฃ๐๐๏ทฏ๏ทฏ Replacing y with n g(n) = ๐โ1 , ๐๐ ๐ ๐๐ ๐๐๐๏ทฎ๐+1, ๐๐ ๐ ๐๐ ๐๐ฃ๐๐๏ทฏ๏ทฏ = f(n) โด Inverse of f is f itself

Miscellaneous

Misc 1
Deleted for CBSE Board 2022 Exams

Misc 2 Deleted for CBSE Board 2022 Exams You are here

Misc 3 Important Deleted for CBSE Board 2022 Exams

Misc. 4 Important

Misc 5

Misc 6 Deleted for CBSE Board 2022 Exams

Misc 7 Deleted for CBSE Board 2022 Exams

Misc. 8 Important

Misc 9 Important Deleted for CBSE Board 2022 Exams

Misc 10 Important

Misc 11 Important Deleted for CBSE Board 2022 Exams

Misc 12 Deleted for CBSE Board 2022 Exams

Misc 13 Important Deleted for CBSE Board 2022 Exams

Misc 14 Important Deleted for CBSE Board 2022 Exams

Misc 15

Misc 16 Important

Misc 17 Important

Misc 18 Deleted for CBSE Board 2022 Exams

Misc, 19 Important Deleted for CBSE Board 2022 Exams

Chapter 1 Class 12 Relation and Functions (Term 1)

Serial order wise

About the Author

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 10 years. He provides courses for Maths and Science at Teachoo.