Q1406: If $f:R^{+}\rightarrow R$ is defined as $f(x) = \log_{a} x$ ...

Class CBSE Class 12 Mathematics Relations and Functions Q #1406

KNOWLEDGE BASED

REMEMBER

3 Marks 2025 AISSCE(Board Exam) SA

If $f:R^{+}\rightarrow R$ is defined as $f(x) = \log_{a} x$ ($a > 0$ and $a\ne1$), prove that f is a bijection. ($R^{+}$ is a set of all positive real numbers.)

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AI Tutor Explanation

Detailed Solution

Step 1: Prove f is injective (one-to-one)

To prove that $f$ is injective, we need to show that if $f(x_1) = f(x_2)$, then $x_1 = x_2$ for all $x_1, x_2 \in R^{+}$.

Step 2: Assume $f(x_1) = f(x_2)$

Let $f(x_1) = f(x_2)$. Then, $\log_{a} x_1 = \log_{a} x_2$.

Step 3: Use properties of logarithms

Since the logarithm function is one-to-one, we can say that if $\log_{a} x_1 = \log_{a} x_2$, then $x_1 = x_2$.

Step 4: Conclude injectivity

Therefore, $f$ is injective (one-to-one).

Step 5: Prove f is surjective (onto)

To prove that $f$ is surjective, we need to show that for every $y \in R$, there exists an $x \in R^{+}$ such that $f(x) = y$.

Step 6: Let $y \in R$

Let $y \in R$. We want to find an $x \in R^{+}$ such that $f(x) = y$, i.e., $\log_{a} x = y$.

Step 7: Solve for x

We can rewrite the equation $\log_{a} x = y$ as $x = a^y$.

Step 8: Verify x is in the domain

Since $a > 0$, $a^y > 0$ for all $y \in R$. Thus, $x = a^y \in R^{+}$.

Step 9: Conclude surjectivity

Therefore, for every $y \in R$, there exists an $x = a^y \in R^{+}$ such that $f(x) = y$. Hence, $f$ is surjective (onto).

Step 10: Conclude bijectivity

Since $f$ is both injective and surjective, $f$ is a bijection.

Final Answer: f is a bijection

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Pedagogical Audit

Bloom's Analysis: This is an REMEMBER question because it requires recalling the definitions of injective, surjective, and bijective functions, as well as the properties of logarithms.

Knowledge Dimension: CONCEPTUAL

Justification: The question requires understanding the concepts of injective, surjective, and bijective functions, and how they relate to the properties of logarithmic functions.

Syllabus Audit: In the context of CBSE Class 12, this is classified as KNOWLEDGE. It directly tests the student's understanding of functions and their properties, which is a core topic in the Relations and Functions chapter.