Given z = x^2 + ((y − 3)^2) − 9.i) Determine the domain and range of the function.ii) Sketch the level curve of the function for z = 0 and z = 7

i) The domain and range of the function:

The domain of the function is all real numbers for x and y. This is because there are no restrictions on the values that x and y can take in the function z = x^2 + ((y − 3)^2) − 9.

The range of the function is all real numbers greater than or equal to -9. This is because the minimum value of x^2 and (y-3)^2 is 0 (which occurs when x=0 and y=3 respectively), so the minimum value of the function is -9.

ii) Sketching the level curve of the function for z = 0 and z = 7:

For z = 0, we have 0 = x^2 + ((y − 3)^2) − 9. Rearranging, we get x^2 + (y-3)^2 = 9, which is the equation of a circle with center at (0,3) and radius 3.

For z = 7, we have 7 = x^2 + ((y − 3)^2) − 9. Rearranging, we get x^2 + (y-3)^2 = 16, which is the equation of a circle with center at (0,3) and radius 4.

So, the level curves for z = 0 and z = 7 are circles with the same center at (0,3) but with different radii (3 and 4 respectively).