Q. 23.7( 27 Votes )

Let f : W W be defined as f (n) = n – 1, if n 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.

Answer :

It is given that f : W W be defined as

f(n) =


Let f(n) = f(m)


We can see that if n is odd and m is even, then we will have n -1 = m +1.


n – m = 2


this is impossible


Similarly, the possibility of n being even and m being odd can also be ignored under a similar argument.


Therefore, both n and m must be either odd or even.


Now, if both n and m are odd, then we get:


f(n) = f(m)


n -1 = m -1


n = m


Again, if both n and m are even, the we get:


f(n) = f(m)


n +1 = m + 1


n = m


f is one – one.


Now, it is clear that any odd number 2r + 1 in co-domain N is the image of 2r in domain N and any even number 2r in co – domain N is the image of 2r +1 in domain N.


f is onto.


f is an invertible function.


Now, let us define g : W W be defined as



Now, when n is odd:


gof(n) = g(f(n)) = g (n-1) = n -1 +1 = n (when n is odd, then n-1 is even)


And when n is even:


gof(n) = g(f(n)) = g (n+1) = n +1 -1 = n (when n is even, then n+1 is odd)


Similarly, when m is odd:


fog(m) = f(g(m)) = f (m-1) = m -1 +1 = m


And when n is even:


fog(m) = f(g(m)) = f (m+1) = m +1 -1 = m


Therefore, gof = IW and fog = IW


Therefore, f is invertible and the inverse of f is given by f-1 = g, which is the same as f.


Thus, the inverse of f is f itself.


Rate this question :

How useful is this solution?
We strive to provide quality solutions. Please rate us to serve you better.
Related Videos
Functions - 01Functions - 01Functions - 0152 mins
Different kind of mappingsDifferent kind of mappingsDifferent kind of mappings58 mins
Some standard real functionsSome standard real functionsSome standard real functions61 mins
Quick Revision of Types of RelationsQuick Revision of Types of RelationsQuick Revision of Types of Relations59 mins
Range of FunctionsRange of FunctionsRange of Functions58 mins
Battle of Graphs | various functions & their GraphsBattle of Graphs | various functions & their GraphsBattle of Graphs | various functions & their Graphs48 mins
Functions - 09Functions - 09Functions - 0947 mins
Quick Recap lecture of important graphs & functionsQuick Recap lecture of important graphs & functionsQuick Recap lecture of important graphs & functions58 mins
Range of Quadratic/quadratic & linear/Linear functionsRange of Quadratic/quadratic & linear/Linear functionsRange of Quadratic/quadratic & linear/Linear functions45 mins
Understand Fractional part and Mod Function in 1 hourUnderstand Fractional part and Mod Function in 1 hourUnderstand Fractional part and Mod Function in 1 hour61 mins
Try our Mini CourseMaster Important Topics in 7 DaysLearn from IITians, NITians, Doctors & Academic Experts
Dedicated counsellor for each student
24X7 Doubt Resolution
Daily Report Card
Detailed Performance Evaluation
caricature
view all courses