GUJCET 2025 Math Paper

Concept:A function is one-one if different inputs always give different outputs.A function is onto if every element in the codomain has at least one preimage in the domain.Explanation:Assume $f(x_1)=f(x_2)$.Then $x_1^3=x_2^3$.Taking cube roots gives $x_1=x_2$, so $f$ is one-one.Also, $x^3$ is strictly increasing on $\mathbb{R}$.Now take any $y \in \mathbb{R}$.Choose $x=\sqrt[3]{y}$, which is real for every real $y$.Then $f(x)=(\sqrt[3]{y})^3=y$.Thus every $y \in \mathbb{R}$ has a preimage in $\mathbb{R}$.So $f$ is onto.Since $f$ is both one-one and onto, it is bijective.Answer:$f(x)=x^3$ is one-one and onto.Correct option: D.