Question

Let \( f : \mathbb{R} \to \mathbb{R} \) be defined as \( f(x) = 10 - x^2 \), then:

• A. \( f \) is one-one and onto.
• B. \( f \) is one-one but not onto.
• C. \( f \) is neither one-one nor onto.
• D. \( f \) is onto but not one-one.

Answer 1

The Correct Option is D

To assess the properties of the function \( f(x) = 10 - x^2 \), we will examine its injectivity and surjectivity.

Injectivity (One-One):

A function is injective if distinct inputs yield distinct outputs. Assuming \( f(a) = f(b) \):

\(10 - a^2 = 10 - b^2\)

Simplifying yields:

\(a^2 = b^2\)

This leads to:

\(a = b\) or \(a = -b\)

Since \(a = -b\) is a possible outcome in addition to \(a = b\), \(f(x)\) is not injective.

Surjectivity (Onto):

A function is surjective if for every \(y \in \mathbb{R}\), there exists at least one \(x \in \mathbb{R}\) such that \(f(x) = y\).

For \(f(x) = 10 - x^2\), we rearrange to solve for \(x^2\):

\(x^2 = 10 - y\)

This requires \(y \leq 10\), as \(x^2\) must be non-negative (\(x^2 \geq 0\)). Any \(y\) value less than or equal to 10 can be attained by a real \(x\), for instance, \(x = \sqrt{10 - y}\) or \(x = -\sqrt{10 - y}\).

Therefore, \(f(x)\) maps to all \(y \leq 10\), confirming that \(f(x)\) is surjective.

Conclusion:

\(f(x)\) is surjective but not injective. The function is onto but not one-to-one.