Specify the region over which the following function f(x) is increasing and the region where it is decreasing. f(x)=(1) / |x|.
Give your answer in open intervals.
To determine the regions over which the function f(x) = |x| is increasing or decreasing, we need to examine the behavior of the function around the critical points. In this case, the critical points occur at x = 0, where the absolute value sign changes the behavior of the function.
1. First, let's consider the interval x < 0. For any negative value of x, the absolute value |x| equals -x. So, in this interval, the function f(x) = 1 / |x| can be written as f(x) = 1 / (-x) = -1/x. As x approaches 0 from the left, the function approaches negative infinity. Thus, the function f(x) is decreasing in the interval (-∞, 0).
2. Next, let's examine the interval x > 0. For any positive value of x, the absolute value |x| remains positive, so f(x) = 1 / |x| is simply f(x) = 1 / x. As x approaches 0 from the right, the function approaches positive infinity. Therefore, f(x) is increasing in the interval (0, ∞).
Hence, the function f(x) = 1 / |x| is decreasing in the interval (-∞, 0) and increasing in the interval (0, ∞).