Question

Given below are two statements: Statement I: The function \(f:\mathbb{R}\to\mathbb{R}\) defined by \[ f(x)=\frac{x}{1+|x|} \] is one-one. Statement II: The function \(f:\mathbb{R}\to\mathbb{R}\) defined by \[ f(x)=\frac{x^2+4x-30}{x^2-8x+18} \] is many-one. In the light of the above statements, choose the correct answer.

• A. Statement I is true but Statement II is false
• B. Both Statement I and Statement II are true
• C. Statement I is false but Statement II is true
• D. Both Statement I and Statement II are false

Hint:

A function defined on \(\mathbb{R}\) is one-one if it is strictly monotonic throughout its domain.

Answer 1

The Correct Option is B

To determine the accuracy of the given statements, let's analyze each function and check their properties.

Analyzing Statement I:

The function \( f(x) = \frac{x}{1+|x|} \) is defined for all \(x \in \mathbb{R}\). We need to check if the function is one-one. A function is one-one (injective) if different inputs produce different outputs, i.e., \(f(x_1) = f(x_2)\) implies \(x_1 = x_2\).

Suppose, \(f(x_1) = f(x_2)\). Then,

\(\frac{x_1}{1+|x_1|} = \frac{x_2}{1+|x_2|}\)

Cross-multiplying gives:

\(x_1(1+|x_2|) = x_2(1+|x_1|)\)

Simplifying, we get:

\(x_1 + x_1|x_2| = x_2 + x_2|x_1|\)

This can be checked further for different cases, but generally solving yields \(x_1 = x_2\). Therefore, the function is one-one. So, Statement I is true.

Analyzing Statement II:

The function \(f(x) = \frac{x^2+4x-30}{x^2-8x+18}\) is defined for all \(x \in \mathbb{R}\) where the denominator does not become zero. We need to check if this function is many-one.

First, let's find the zeros of the denominator:

\(x^2 - 8x + 18 = 0\)

By using the quadratic formula, \(x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\), we can find the roots where:

\(x = \frac{8 \pm \sqrt{64 - 72}}{2}\)

\(x = \frac{8 \pm \sqrt{-8}}{2}\)

These are non-real numbers, so the function is defined for all real \(x\).

To check if it is many-one, we can examine if there exists at least one \(y\) that has multiple \(x\) values such that \(f(x_1) = f(x_2) = y\). Simplifying such expressions usually yields possible distinct values of \(x\) with the same \(f(x)\) value.

Upon analyzing, it can be deduced that multiple values of \(x\) can map to the same value of the output, thus the function is many-one. Hence, Statement II is true.

Conclusion:

From the analysis above, both Statement I and Statement II are true.

Therefore, the correct answer is: Both Statement I and Statement II are true.