Show that the function f : Z → Z : f (x) = x3 is one-one and into.

1.	Show that the function f : Z → Z : f (x) = x3 is one-one and into.
Answer» To prove: function is one-one and into Given: f : Z → Z : f (x) = x³ Solution: We have, f(x) = x³ For, f(x₁) = f(x₂) ⇒ x₁³ = x₂³ ⇒ x₁ = x₂ When, f(x₁) = f(x₂) then x₁ = x₂ ∴ f(x) is one-one f(x) = x³ Let f(x) = y such that \(y\in Z\) ⇒ y = x³ ^{\(\Rightarrow x=\sqrt[3]{y}\)} If y = 2, as \(y\in Z\) Then we will get an irrational value of x, but \(x\in Z\) Hence f(x) is into Hence Proved

Answer»

To prove: function is one-one and into

Given: f : Z → Z : f (x) = x³

Solution: We have,

f(x) = x³

For, f(x₁) = f(x₂)

⇒ x₁³ = x₂³

⇒ x₁ = x₂

When, f(x₁) = f(x₂) then x₁ = x₂

∴ f(x) is one-one

f(x) = x³

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

⇒ y = x³

^{\(\Rightarrow x=\sqrt[3]{y}\)}

If y = 2, as \(y\in Z\)

Then we will get an irrational value of x, but \(x\in Z\)

Hence f(x) is into

Hence Proved

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