Class 12 Relations And Functions · Very Long Answer5 Markstypes_of_functions

Class 12 Maths · Relations And Functions · Very Long Answer #191

Let W={0,1,2,…}. Define f:W→W by f(n)=n−1 for odd n and n+1 for even n. Prove invertibility and find f⁻¹.

✅ Step-by-Step Solution

Final Answer: f⁻¹=f.

An odd n is at least 1, so n−1∈W; for even n, n+1∈W. Thus f is well defined.
For even n, f(n)=n+1 is odd and f(f(n))=(n+1)−1=n.
For odd n, f(n)=n−1 is even and f(f(n))=(n−1)+1=n.
The identity f∘f=id implies injectivity (apply f to equal images) and surjectivity (n is the image of f(n)).
Therefore f is invertible and f⁻¹(n)=n−1 for odd n, n+1 for even n; its inverse is itself.

[1 mark] An odd n is at least 1, so n−1∈W; for even n, n+1∈W. Thus f is well defined.
[1 mark] For even n, f(n)=n+1 is odd and f(f(n))=(n+1)−1=n.
[1 mark] For odd n, f(n)=n−1 is even and f(f(n))=(n−1)+1=n.
[1 mark] The identity f∘f=id implies injectivity (apply f to equal images) and surjectivity (n is the image of f(n)).
[1 mark] Therefore f is invertible and f⁻¹(n)=n−1 for odd n, n+1 for even n; its inverse is itself.

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