If f(x) = -|x - h| + k, what is the range of the function?

If f(x) = -|x - h| + k, what is the range of the function?

• A. y >= k
• B. All real numbers
• C. y < k
• D. y <= k

Answer: (D)

A shift-and-reflect shape: the absolute value is always nonnegative, and placing a minus in front reflects it downward. So f(x) = k − |x − h| is never greater than k, with the highest point at x = h where f(h) = k. As x moves away from h, |x − h| grows and f(x) decreases without bound, meaning the values go down to negative infinity. This gives a range of all y such that y ≤ k, including k itself. So the correct description is y ≤ k.

A shift-and-reflect shape: the absolute value is always nonnegative, and placing a minus in front reflects it downward. So f(x) = k − |x − h| is never greater than k, with the highest point at x = h where f(h) = k. As x moves away from h, |x − h| grows and f(x) decreases without bound, meaning the values go down to negative infinity. This gives a range of all y such that y ≤ k, including k itself. So the correct description is y ≤ k.