Final answer:
The function h(x, y) = xy² + 3x - 4y has a domain and range both consisting of all real numbers, as there are no constraints on the values that x and y can take, and the function's output is not limited to just positive or negative values.
Step-by-step explanation:
The student is asking about the domain and range of the multivariable function h(x, y) = xy² + 3x - 4y. Let's analyze the function:
- The domain of a function refers to the set of all possible input values (x and y in this case) that the function can accept.
- The range of a function is the set of all possible output values it can produce.
Observing the given function h(x, y) = xy² + 3x - 4y, we can see there are no denominators or even roots, meaning the domain includes all real numbers for x and y without any restriction.
Considering the range, we cannot determine any specific limitations just by looking at the expression. Since the function includes terms that can generate both positive and negative values depending on the choice of x and y (for instance, if x and y are both positive or both negative the first term xy² is positive, whereas if x is negative and y is positive, the first term is negative), the output can indeed be any real number. Therefore, the range also consists of all real numbers.
The correct answer is, hence, a) Domain: All real numbers, Range: All real numbers.