Final answer:
The domain of h(x) = -2| x | is all real numbers, the range is all non-positive real numbers, and the function is negative for all non-zero values of x and zero at x=0. The domain and range of a relation are defined as the sets of all the x-coordinates and y-coordinates of the ordered pairs, respectively. For instance, if the relation is, R = {(1, 2), (2, 2), (3, 3), (4, 3)}, then: Domain = the set of all x-coordinates = {1, 2, 3, 4} Range = the set of all y-coordinates = {2, 3}
Step-by-step explanation:
When examining the function h(x) = -2| x |, which is a reflection of f(x) = 2| x |, we need to describe the domain, range, and the intervals over which the function is positive or negative.
The domain of both f(x) and h(x) is all real numbers, since an absolute value function can take any real number as an input. This can be expressed as x is a real number.
The range of h(x), however, is different from f(x) due to the reflection over the x-axis. The range of h(x) is all non-positive real numbers, which means the function can have values that are zero or negative. This is expressed as y ≤ 0.
As for the intervals where the function is positive or negative, since h(x) is a reflection of the absolute value function over the x-axis, it will be negative wherever the original function f(x) is positive. Thus, h(x) is negative for all non-zero values of x, which can be stated as the intervals (-∞, 0) and (0, +∞). The function is zero when x is exactly zero, which is the single point interval [0].
The domain is defined as the entire set of values possible for independent variables. The Range is found after substituting the possible x- values to find the y-values.