Prove that the function f: N → N : f (x)=3x is one-one and into.

1.	Prove that the function f: N → N : f (x)=3x is one-one and into.
Answer» To prove: function is one-one and into Given: f: N → N : f(x)= 3x We have, f(x) = 3x For, f(x₁) = f(x₂) ⇒ 3x₁ = 3x₂ ⇒ x₁ = x₂ When, f(x₁) = f(x₂) then x₁ = x₂ ∴ f(x) is one-one f(x) = 3x Let f(x) = y such that \(y\in N\) ⇒ y = 3x ⇒ \(x=\frac{y}{3}\) If y = 1, ⇒ \(x=\frac{1}{3}\) But as per question \(x\in E\), hence x can not be \(\frac{1}{3}\) Hence f(x) is into Hence Proved

Answer»

To prove: function is one-one and into

Given: f: N → N : f(x)= 3x

We have,

f(x) = 3x

For, f(x₁) = f(x₂)

⇒ 3x₁ = 3x₂

⇒ x₁ = x₂

When, f(x₁) = f(x₂) then x₁ = x₂

∴ f(x) is one-one

f(x) = 3x

Let f(x) = y such that \(y\in N\)

⇒ y = 3x

⇒ \(x=\frac{y}{3}\)

If y = 1,

⇒ \(x=\frac{1}{3}\)

But as per question \(x\in E\), hence x can not be \(\frac{1}{3}\)

Hence f(x) is into

Hence Proved

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