Question

Let f(x, y,z) = e √ z−x 2−y 2 . (a) Evaluate f(2,−1,6) (b) Find the domain of f (c) Find the range of f

Answer 1

$$f(x,y,z)=e^{\sqrt{z-x^2-y^2}}$$

a)

$$f(2,-1,6)=e^{\sqrt{6-(2)^2-(-1)^2}}=e$$

b)

$$z-x^2-y^2\geq 0$$

Domain: $z-x^2-y^2\geq 0\ or\ z\geq x^2+y^2.$ The set of points that lie on the elliptic paraboloid $z=x^2+y^2$ or inside it.

c)

$$\sqrt{z-x^2-y^2}\geq0=>e^{\sqrt{z-x^2-y^2}}\geq 1$$

Range: $[1,\infin).$