Mathematics

Problem

Consider the function

f : \mathbb R \to \mathbb R

defined by

f(x)=x^{3}+4

. a. (i) Prove that

f

is one–one and onto. b. (ii) Hence state the domain and the range of

f

. c. (iii) Without finding an explicit formula for

f^{-1}

, write the domain and the range of

f^{-1}

and give reasons for your answer.

Intuitive explanation

The key to understanding this cubic function is recognizing that $f(x) = x^3 + 4$ is just a vertical shift of the basic cubic $x^3$ . Since $x^3$ is strictly increasing and one-to-one everywhere, adding a constant doesn't change that behavior—it only slides the entire curve up by 4 units. This means $f$ is invertible and preserves all the monotonicity and symmetry properties of the parent cubic.

Intuitive explanation

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