Question

Let \( X \) and \( Y \) be subsets of \( \mathbb{R} \). If \( f : X \rightarrow Y \) given by \( f(x) = -8(x + 5)^2 \) is one-to-one, then the codomain \( Y \) is:

• A. \( (-\infty, 0] \)
• B. \( (-\infty, -5] \)
• C. \( (-\infty, -5) \)
• D. \( [0, -\infty) \)
• E. \( (-\infty, \infty) \)

Hint:

For quadratic functions, the range is determined by the square of the expression inside, and the constant factor applied outside.

Answer 1

The Correct Option is A

Since \( f(x) = -8(x + 5)^2 \), we know that \( (x + 5)^2 \) is always non-negative, and so \( f(x) \leq 0 \) for all values of \( x \). 
Therefore, the function maps to values in the range \( (-\infty, 0] \).