Final answer:
The function f(x) = -1/|x-4| is increasing for x < 4 and decreasing for x > 4, as determined by analyzing the behavior of the function with respect to the absolute value in the denominator.
Step-by-step explanation:
The student has asked about the behavior of the function f(x) = -1/|x-4|. We can analyze the function's behavior by determining where the function is increasing and decreasing based on the value of x. Since we have an absolute value in the denominator, we can consider two cases: one when x is greater than 4 and the other when x is less than 4.
Case 1: x > 4
In this case, the absolute value |x - 4| is just x - 4 since x is already positive with respect to 4. As x increases, x - 4 also increases, which makes the absolute value larger and consequently the whole fraction smaller (more negative). Therefore, f(x) is decreasing for x > 4.
Case 2: x < 4
When x is less than 4, |x - 4| is equivalent to 4 - x since we have to flip the sign to make it positive. As x increases toward 4, 4 - x becomes smaller, which makes the absolute value smaller, thus the whole fraction larger (less negative). So, f(x) is increasing for x < 4. Remember that as x approaches 4 from either side, the function goes to negative infinity because the denominator approaches 0.