Misc 2 - Chapter 1 Class 12 Relation and Functions (Term 1)
Last updated at May 29, 2018 by
Last updated at May 29, 2018 by
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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
Finding Inverse
Inverse of a function
How to check if function has inverse?
Example 22 Deleted for CBSE Board 2022 Exams
Ex 1.3, 5 (i) Deleted for CBSE Board 2022 Exams
How to find Inverse?
Example 28 (a) Deleted for CBSE Board 2022 Exams
Misc 11 (i) Important Deleted for CBSE Board 2022 Exams
Ex 1.3, 11 Deleted for CBSE Board 2022 Exams
Example 27 Important Deleted for CBSE Board 2022 Exams
Misc 1 Deleted for CBSE Board 2022 Exams
Ex 1.3 , 6 Deleted for CBSE Board 2022 Exams
Ex 1.3, 14 (MCQ) Important Deleted for CBSE Board 2022 Exams
Example 23 Important Deleted for CBSE Board 2022 Exams
Misc 2 Deleted for CBSE Board 2022 Exams You are here
Ex 1.3 , 4 Deleted for CBSE Board 2022 Exams
Example 24 Deleted for CBSE Board 2022 Exams
Ex 1.3 , 8 Important Deleted for CBSE Board 2022 Exams
Example 25 Important Deleted for CBSE Board 2022 Exams
Ex 1.3 , 9 Important Deleted for CBSE Board 2022 Exams
Finding Inverse
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