Final answer:
The domain of the function √(1 + x - y²) is the region graphically represented on or below the parabola y² = 1 + x, and the range is [0, ∞) as the square root function outputs only non-negative values with no upper limit.
Step-by-step explanation:
To find and sketch the domain of the function f(x, y) = √(1 + x - y²), you must understand that the domain consists of all the (x, y) pairs for which the value inside the square root is non-negative because the square root function is only defined for non-negative numbers. Therefore, the domain is determined by the condition 1 + x - y² ≥ 0. Rearranging, we get y² ≤ 1 + x.
Graphically, this inequality represents the region on or below the parabola y² = 1 + x, which opens towards the positive x-direction. To sketch the domain, simply plot this parabola and shade the entire region to its right.
The range of f(x, y) is the set of all possible output values. Since f(x, y) represents a square root, its smallest value is 0, occurring when x and y are such that 1 + x - y² = 0. There is no upper bound since x can increase indefinitely, making f(x, y) as large as desired. Hence, the range of f(x, y) is [0, ∞).