The graphs for the following equations are attached below.
Here are the domains and ranges of each function:
a. g(x) = -1/4 |x|
Domain: All real numbers. Since the absolute value function accepts any real number as input, there are no restrictions on the values of x.
Range: All non-negative real numbers (values greater than or equal to 0). The absolute value always results in a non-negative output, regardless of the sign of the input.
b. h(x) = 3.5 |x|
Domain: All real numbers. Same reasoning as for g(x).
Range: All real numbers greater than or equal to 0. The absolute value multiplied by 3.5 simply scales the output upwards, but it remains non-negative.
c. p(x) = -5 |x|
Domain: All real numbers. Same reasoning as for g(x).
Range: All non-positive real numbers (values less than or equal to 0). Multiplying the absolute value by -5 flips the sign of the output, making all values non-positive.
d. d(x) = 1/3 |x|
Domain: All real numbers. Same reasoning as for g(x).
Range: All real numbers greater than or equal to 0. Similar to h(x), the absolute value multiplied by 1/3 scales the output down but keeps it non-negative.