Question

What is the domain of the function f(x,y)=x^8 y^1/3 −5lnx ?

(A) {(x,y):x>0,y≥0}
(B) {(x,y):x≥0,y≥0}
(C) {(x,y):y≥0}
(D) {(x,y):x≥0}
(E) {(x,y):y>0}
(F) {(x,y):x>0}
(G) {(x,y):x≥0,y>0}
(H) {(x,y):x>0,y>0}

Answer 1

The domain of the function is defined by the conditions x>0 because of the natural logarithm, and y≥0 because we're taking the cube root of y. Thus, the domain is all pairs (x,y) where x is greater than zero and y is greater than or equal to zero.

Step-by-step explanation:

The domain of the function f(x,y)=x^8 y^1/3 − 5lnx can be determined by looking at the restrictions imposed by the two parts of the function. The term x^8 is defined for all real x since any real number raised to the 8th power is real. However, the term y^1/3 only makes sense for y≥0 because we are taking the cube root of y. Most critically, the term -5lnx defines the main restriction because the natural logarithm function is only defined for x>0. Therefore, x must be greater than 0 to avoid an undefined expression. Combining these restrictions, we conclude that the domain of the function is {(x,y): x>0, y≥0}, which corresponds to option (A).