Suppose that z(x,y) may be defined implicitly by {x}^2 - {y}^2- {z}^2 = 16, enter inequalities for the domain (x and y values) and the range (z values).
To find the inequalities for the domain (x and y values) and the range (z values) of the implicitly defined function z(x, y), we need to analyze the equation {x}^2 - {y}^2 - {z}^2 = 16.
For the domain, we need to determine the valid values for x and y. The equation implies that any combination of x and y values that satisfy the equation will be in the domain. However, since there are no explicit limitations on x and y within the equation, we can assume that they can take any real value. Therefore, the domain can be represented by the inequalities:
-Domain inequality for x: -∞ < x < +∞
-Domain inequality for y: -∞ < y < +∞
For the range, we need to find the valid values for z. Since z is defined implicitly in terms of x and y, we can solve the equation for z. Rearranging the equation, we have:
{z}^2 = {x}^2 - {y}^2 - 16
Now, since the right-hand side of this equation can take any value, z can also take any real value. Therefore, the range of the function z(x, y) is given by the inequality:
-∞ < z < +∞
Hence, the inequalities for the domain are -∞ < x < +∞ and -∞ < y < +∞, and the inequality for the range is -∞ < z < +∞.