Question

Let \( R: \mathbb{R} \to \mathbb{R} \) be defined as \( f(x) = x^2 + 1 \), find \( f^{-1}(-5) \):

• A. \( \emptyset \)
• B. \( \{ -5 \} \)
• C. \( \{ 5 \} \)
• D. \( \{ -5, 5 \} \)

Hint:

For functions where the range is restricted, ensure that the inverse function's output is valid within the domain.

Answer 1

The Correct Option is A

Step 1: Inverse of the function.
We are asked to find the inverse of the function for \( f(x) = x^2 + 1 \). The inverse function will not produce any negative values, as \( f(x) \) is always greater than or equal to 1.
Step 2: Conclusion.
Thus, \( f^{-1}(-5) = \emptyset \).
Final Answer: \[ \boxed{\emptyset} \]