Question

Let f : R `->` R be a function defined by f(x) = x³ + 4, then f is ______.

Options

• injective

• surjective

• bijective

• none of these

Answer 1

Let f : R `->` R be a function defined by f(x) = x³ + 4, then f is bijective.

Explanation:

Let `f(x_1) = f(x_2) for x_1, x_2` ∈ R

⇒ `x_1^3 + 4 = x_2^3 + 4`

⇒ `x_1^3 - x_2^3` = 0

⇒ `(x_1 + x_2)(x_1^2 + x_2^2 + x_1x_2)` = 0

⇒ `(x_1  -x_2) ((x_1 + x_2/2)^2 + 3/4 x^2)` = 0

x₁ – x₂ = 0 ⇒ x₁ = x₂ 

∴ f is one-one.

Let k ∈ R.

f(x) = k ⇒ x³ + 4 = k ⇒ x = `(k - 4)^(1/3)` ∈ R

∴ f is onto