Question

If f: R `rightarrow` R be a function defined by f(x) = 4x³ – 7. Then ______.

Options

• f is one-one -into

• f is many-one - into

• f is many-one onto

• f is bijective

Answer 1

If f: R `rightarrow` R be a function defined by f(x) = 4x³ – 7. Then f is bijective.

Explanation:

We have f(x) = 4x³ – 7, x ∈ R.

f is one-one.

Let x₁, x₂ ∈ R and f(x₁) = f(x₂).

`\implies 4x_1^3 - 7 = 4x_2^3 - 7`

`\implies 4x_1^3 = 4x_2^3`

 `\implies x_1^3 = x_2^3`

`\implies x_1^3 - x_2^3` = 0.

`\implies (x_1 - x_2)(x_1^2 + x_1x_2 + x_2^2)` = 0.

`\implies (x_1 - x_2)[(x_1 + x_2/2)^2 + (3x_2^2)/4]` = 0.

`\implies` x₁ – x₂ = 0, because the other factor is non-zero.

`\implies` x₁ = x₂    

∴ f is one-one.

f is onto.

Let k ∈ R any real number.

f(x) = k

`\implies` 4x³ – 7 = k

`\implies x = ((k + 7)/4)^(1//3)`

Now `(k + 7/4)^(1//3) ∈ R`, because k ∈ R and 

`f[((k + 7)/4)^(1//3)] = 4[((k + 7)/4) ^(1//3)]^-3 - 7`

= `4((k + 7)/4) - 7` = k

∴ k is the image of `((k + 7)/4)^(1//3)`

∴ f is onto.

∴ f is a bijective function.