For f(x) = √(x − h) + k, the range is described by which inequality?

For f(x) = √(x − h) + k, the range is described by which inequality?

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

Answer: (C)

The range of a square root function after shifts is determined by the fact that the square root output is always nonnegative. Here, sqrt(x − h) is at least 0 for all x in its domain (x ≥ h). Adding k shifts the graph up by k, so the smallest possible value of the function is k, achieved when x = h. As x grows larger, sqrt(x − h) grows without bound, so the function values go up toward infinity. Therefore, the range is all real numbers y such that y ≥ k. The option describing all real numbers would be incorrect because you can’t get values below k from this function.

The range of a square root function after shifts is determined by the fact that the square root output is always nonnegative. Here, sqrt(x − h) is at least 0 for all x in its domain (x ≥ h). Adding k shifts the graph up by k, so the smallest possible value of the function is k, achieved when x = h. As x grows larger, sqrt(x − h) grows without bound, so the function values go up toward infinity. Therefore, the range is all real numbers y such that y ≥ k. The option describing all real numbers would be incorrect because you can’t get values below k from this function.