Mark (√) against the correct answer in the following: f : N → N :

1.	Mark (√) against the correct answer in the following: f : N → N : f(x) = 2x is A. one - one and onto B. one - one and into C. many - one and onto D. many - one and into
Answer» Correct Answer is (B) one - one and into f(x) = 2x For One - One f(x₁) = 2x₁ f(x₂) = 2x₂ put f(x₁) = f(x₂) we get 2x₁ = 2x₂ Hence, if f(x₁) = f(x₂), x₁ = x₂ Function f is one - one For Onto f(x) = 2x let f(x) = y, such that y∈N 2x = y \(\Rightarrow x=\frac{y}{2}\) If y = 1 \(x=\frac{1}{2}=0.5\) which is not possible as x∈N Hence, f is not onto., f is into Hence, option b is correct

Mark (√) against the correct answer in the following: f : N → N : f(x) = 2x is A. one - one and onto B. one - one and into C. many - one and onto D. many - one and into

Answer»

Correct Answer is (B) one - one and into

f(x) = 2x

For One - One

f(x₁) = 2x₁

f(x₂) = 2x₂

put f(x₁) = f(x₂) we get

2x₁ = 2x₂

Hence, if f(x₁) = f(x₂), x₁ = x₂

Function f is one - one

For Onto

f(x) = 2x

let f(x) = y, such that y∈N

2x = y

\(\Rightarrow x=\frac{y}{2}\)

If y = 1

\(x=\frac{1}{2}=0.5\)

which is not possible as x∈N

Hence, f is not onto., f is into

Hence, option b is correct