Advanced MathVery hardRange of a downward-opening parabola
Advanced Math practice question
The function f is defined by f(x) = 8 − (x + 1)². Which of the following describes all possible values of f(x)?
- A. f(x) ≤ 8Correct
- B. f(x) ≥ 8
- C. f(x) ≤ −1
- D. All real numbers
Answer: A. f(x) ≤ 8
(x + 1)² is never negative, so 8 − (x + 1)² is never more than 8, and it equals 8 when x = −1. The range is f(x) ≤ 8.
Why the other answers are wrong
- B. f(x) ≥ 8
- Subtracting a square can only bring the value DOWN from 8, never above it.
- C. f(x) ≤ −1
- −1 is the x-coordinate of the vertex, not a bound on f(x). The maximum value is 8.
- D. All real numbers
- A parabola that opens downward has a maximum, so not every real number is reached.
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