For f(x) = -√(x − h) + k, which statement describes its range?

For f(x) = -√(x − h) + k, which statement describes its range?

• A. y ≥ k
• B. y ≤ k
• C. All real numbers
• D. y ≠ k

Answer: (B)

A square root function has range values that start at zero and go up. In this expression, the minus sign in front of the square root reflects the graph across the x-axis, so the outputs become less than or equal to zero. Then adding k shifts everything up by k, giving a range of values that are less than or equal to k. At x = h, the square root is zero, so f(h) = k, which is the highest value the function can take. As x grows, the square root grows, and the negative sign drives the values downward without bound, so y can be any number as small as you like, but not larger than k. Therefore the range is all real numbers y with y ≤ k.

A square root function has range values that start at zero and go up. In this expression, the minus sign in front of the square root reflects the graph across the x-axis, so the outputs become less than or equal to zero. Then adding k shifts everything up by k, giving a range of values that are less than or equal to k. At x = h, the square root is zero, so f(h) = k, which is the highest value the function can take. As x grows, the square root grows, and the negative sign drives the values downward without bound, so y can be any number as small as you like, but not larger than k. Therefore the range is all real numbers y with y ≤ k.