Question

The function f: R → R defined as f(x) = x³ is:

Options

• One-on but not onto

• Not one-one but onto

• Neither one-one nor onto

• One-one and onto

Answer 1

One-one and onto

Explanation:

Let f(x₁) = f(x₂) such that x₁x₂ ∈ R

`"x"_1^3 = "x"_2^3`

x₁ = x₂

f is one-one

Let y ∈ R (codomain)

Then for any x, f(x) = y if x³ = y

i.e., x = `"y"^(1/3)` ∈ R (domain)

i.e., every element y ∈ R (codomain) has a preimage `"y"^(1/3)` in R (domain)

f is onto

∴ f is one-one and onto.