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

Question

Philoid · Accepted 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 = x³

If y = 2, as

Then we will get an irrational value of x, but

Hence f(x) is into

Hence Proved