Showing a function is strictly decreasing on different intervals

• k3k3
In summary, the function g is strictly decreasing on the intervals (-∞,0) and [0,∞), but g is not decreasing on ℝ.
k3k3

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.

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.

k3k3 said:

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.
Is the function g or f?
I think this is what you meant to say:
Define a function g on ℝ by g(x)=1/x if x < 0 and g(x)=-x if x≥0.
Prove g is strictly decreasing on the intervals (-∞,0) and [0,∞), but that g is not decreasing on ℝ.
k3k3 said:

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.
What's this about the image being between -1 and 0? For example, let x = -1/2 and y = -1/3. Then 1/x = -2 and 1/y = -3.

Since x < y, and both numbers are negative, then 1/x > 1/y, so g is decreasing. I think this is all you need to say.
k3k3 said:
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.

Yes, I meant how you corrected it. Sorry about that!

Other than the first part, the rest looks good?

I didn't look at anything very closely after the comments I inserted earlier. Here's the rest of your proof.
k3k3 said:
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.
Don't say "it follows that f(x) > f(y) because ..." Just show it like I did in my earlier post. What you have above, IMO, is more wordy than it should be, and less mathematical than it should be.

Also, since you are presumably in a calculus class (you posted this in the Calculus & Beyond section), you can show that g'(x) < 0 for all x in (-∞, 0), and there's a connection between that fact and whether a function is increasing or decreasing.
k3k3 said:
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.
g(.5) = -.5, not -2. On the interval you chose -- [0, .5] -- g is in fact decreasing, so this does not further your argument. If you haven't sketched a graph of the function, you should do so, and this will help you understand why this function isn't decreasing on R even though it is decreasing on each half of its domain.

g(.5) was a typo. I meant g(-.5) since when x < 0, g(x) = 1/x and that is why I said -2. That should show why it is not decreasing on all of R since -2 < 0.

Thank you for the suggestions! I will revise my proof.

1. How do you show that a function is strictly decreasing on a given interval?

To show that a function is strictly decreasing on a given interval, you need to prove that for any two points within the interval, the function value at the first point is greater than the function value at the second point. This can be done by finding the derivative of the function and showing that it is negative on the given interval.

2. Can a function be strictly decreasing on one interval and strictly increasing on another?

Yes, a function can be strictly decreasing on one interval and strictly increasing on another. This occurs when the function has a point of inflection within the given interval, where the slope changes from negative to positive or vice versa.

3. What is the difference between strictly decreasing and non-increasing?

The main difference between strictly decreasing and non-increasing is that a strictly decreasing function must have a negative slope throughout the given interval, while a non-increasing function can have a slope of zero at certain points within the interval.

4. How can you determine if a function is strictly decreasing without using calculus?

If the function is given in a table or graph form, you can determine if it is strictly decreasing by observing the trend of the points. If the y-values decrease as the x-values increase, then the function is strictly decreasing. You can also use the first derivative test, where you plug in values on either side of a critical point to see if the slope changes from positive to negative.

5. Can a function be strictly decreasing on its entire domain?

Yes, a function can be strictly decreasing on its entire domain. This means that for any input value, the function output will always decrease. An example of such a function is f(x) = -x, where the slope is always negative.

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