determine the domain and range for the function f(x,y)=Sq.Rt. 1-x*2-y*2
To determine the domain and range of the function f(x, y) = √(1 - x^2 - y^2), we need to consider the restrictions imposed by the function's formula.
1. Domain:
The domain represents all possible input values for the function f(x, y). In this case, the function involves square roots, which means the expression within the square root must be non-negative (i.e., greater than or equal to zero) to avoid taking the square root of a negative number. Therefore, we can set up the inequality:
1 - x^2 - y^2 ≥ 0
To solve this inequality, let's rearrange it:
-x^2 - y^2 + 1 ≥ 0
Now, recall that x^2 + y^2 represents the equation of a circle centered at the origin with a radius of 1. So, the inequality states that any point (x, y) lying on or inside this circle satisfies the condition. Thus, the domain for f(x, y) is the disc enclosed by the circle.
2. Range:
The range represents all possible output values produced by the function f(x, y). Since the function involves a square root, it can only produce non-negative values. Thus, the range of f(x, y) includes all real numbers greater than or equal to zero.
In summary:
Domain: The disc enclosed by the circle x^2 + y^2 ≤ 1.
Range: All real numbers greater than or equal to zero.