Q2133: Let $f : R \to R$ be a function defined by $f(x) = x^3 + 4$,...

CBSE Class 12 Mathematics Relations and Functions Q #2133

COMPETENCY BASED

APPLY

5 Marks Generic-Curriculum LA

Let $f : R \to R$ be a function defined by $f(x) = x^3 + 4$, then show that $f$ is bijective.

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Detailed Solution

Step 1: Proving Injectivity (One-to-One)

A function is injective if $f(x_1) = f(x_2)$ implies $x_1 = x_2$. Given $f(x) = x^3 + 4$, we set $f(x_1) = f(x_2)$:

$$x_1^3 + 4 = x_2^3 + 4$$ $$x_1^3 = x_2^3$$ $$x_1 = x_2$$

Since $x_1 = x_2$, the function is injective.

Step 2: Proving Surjectivity (Onto)

A function is surjective if for every $y \in R$ (codomain), there exists an $x \in R$ (domain) such that $f(x) = y$.

$$y = x^3 + 4$$ $$x^3 = y - 4$$ $$x = \sqrt[3]{y - 4}$$

Since for every real number $y$, $\sqrt[3]{y - 4}$ is a well-defined real number, the function is surjective.

Step 3: Conclusion

Since the function is both injective and surjective, it is bijective.

Final Answer: The function is bijective as it satisfies both one-to-one and onto conditions.

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Pedagogical Audit

Bloom's Analysis: This is an APPLY question because the student must apply the formal definitions of injectivity and surjectivity to a specific algebraic function.

Knowledge Dimension: PROCEDURAL

Justification: The student follows a standard algorithmic process to verify the properties of a function.

Syllabus Audit: In the context of CBSE Class 12, this is classified as COMPETENCY. This question tests the fundamental understanding of Relations and Functions (Chapter 1), which is a core competency for calculus readiness.

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