1. The problem statement, all variables and given/known data Define a function g on ℝ by f(x)=1/x if x < 0 and f(x)=-x if x≥0. Prove f is strictly decreasing on the intervals (-∞,0) and [0,∞), but that g is not decreasing on ℝ. 2. Relevant equations 3. The attempt at a solution I think I understand what I need to show, but I often have trouble properly showing it. I broke it up into the three cases that the directions seem to indicate. Show g(x)=1/x is strictly decreasing on (-∞,0) Let x and y be any elements in the interval (-∞,0) that satisfy the inequality x < y. Since the function on the interval is just the reciprocal, all of the image is between -1 and 0. For any x < y, it follows that f(x) > f(y) because the larger the magnitude of the element, the further away from 0 the element is and thus the closer its reciprocal is to 0. Show g(x)=-x is strictly decreasing on [0,∞) Since g is a linear function on the interval [0,∞) with a negative slope, this function is strictly decreasing on [0,∞). Show g is not strictly decreasing on ℝ Suppose g is strictly decreasing on ℝ Then g(.5) > g(0)→ -2 > 0 This is not true, so it cannot be decreasing at this portion of the range. Hence, g is not strictly decreasing on ℝ.