Question

Consider the function f :N—N defined by
f(x) = x+1, if x is odd
x-1, if x is even
la) Write f(1) and f(2)
(b) show that f is bijective
(c) Find the inverse of f, if it exists​

Answer 1

Given :  f(x) = x+1, if x is odd  , x-1, if x is even

To find : f(1) and f(2)

Solution:

f(x) = x+1, if x is odd

x-1, if x is even

f(1) = 1 + 1 = 2

f(2) = 2 - 1  = 1

f(1) = 2

f(2) = 1

x₁  & x₂  are odd

f(x₁) = x₁ + 1   f(x₂) = x₂ + 1

x₁ + 1 = x₂ + 1

if x₁  = x₂

x₁  & x₂  are even

f(x₁) = x₁ - 1   f(x₂) = x₂ - 1

x₁ - 1 = x₂ - 1

if x₁  = x₂

x₁  odd  & x₂ even

x₁  + 1 = x₂  - 1

=>  x₂ - x₁ = 2  ( not possible as even - odd can not be 2)

f(x₁) ≠ f(x₂) if x₁ ≠  (x₂ )

f(x) = x + 1    x is odd

y = x + 1    => y is even

y -  1  = x

=> x = y - 1    

Similarly :

x = y - 1  if y is even

    y + 1  if y is odd

f⁻¹(x) = x - 1  if x is even

    x + 1  if x is odd

Hence f⁻¹(x)  = f(x)

Learn more:

मान लीजिए कि समस्त n ∈ N के लिए, ????(n)={ (n+1)/2,यदि ...

https://brainly.in/question/16554601