Question

Consider the functions f(x) = (x - 1)(x + 2), g(x) = sin²(x), and h(x) = x / |x| - 1. For each function, (i) find the natural domain and range; (ii) determine whether the function is odd, even

Answer 1

The natural domain and range of the given functions are determined using their restrictions on variables. The functions are analyzed individually to find their natural domains and ranges.

Step-by-step explanation:

To find the natural domain and range of the given functions, we consider the restrictions on the variables.

Function f(x) = (x - 1)(x + 2):

The natural domain is all real numbers, as there are no restrictions on x. The range can be determined by considering the graph of the function, which is a parabola that opens upward. Therefore, the range is all real numbers greater than or equal to the y-coordinate of the vertex.

Function g(x) = sin²(x):

The natural domain is all real numbers, as the sine function is defined for all real numbers. The range is between 0 and 1, inclusive, as the square of the sine function will always be between 0 and 1.

Function h(x) = x / |x| - 1:

The natural domain is all real numbers except x = 0, as the function is not defined for x = 0 due to the division by zero. The range is all real numbers except y = -1, as the absolute value of x will always be greater than or equal to 0.