- #1

k3k3

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## Homework Statement

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.

## Homework Equations

## 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.