Question

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 1

Step-by-step explanation:

ANSWER

f:W→W

f(x) ={n−1,n=oddn+1,n=even}

When n is odd

f(x

1

)=f(x

2

)

n

1

−1=n

2

−1

n

1

=n

2

When n is even

f(x

1

)=f(x

2

)

n

1

+1=n

2

+1

n

1

=n

2

So, f(x) is one-one

When n is odd

f(x)=n−1

y=n−1

n=y+1

Put n in f(x)

f(x)=y+1−1

f(x)=y

When n is even

f(x)=n+1

y=n+1

n=y−1

Put n in f(x)

f(x)=y−1+1

f(x)=y

So, f(x) is onto

So, the function f(x) is bijective. Hence is invertible.

f(x)=n−1 if n is odd

y=n−1

n=y−1

f

−1

(x)=y−1 if n is odd

f(x)=n+1 if n is even

y=n+1

n=y+1

f

−1

(x)=y+1 if n is even

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