Strictly decreasing

JeffM said:
This problem is a bit tricky because the intervals specified in defining the function are not sufficient to determine the question of the intervals that are strictly decreasing. We need the definition of strictly decreasing, namely

[math]\text {Given } a < b \text { and } a \le x \le b \implies f(x) \in \mathbb R,\\ f(x) \text { is strictly decreasing over } (a, \ b) \iff f(a) > f(x) > f(b).[/math]
This definition must be adjusted in obvious ways for partially or fully closed intervals.

Now the absolute value function is not everywhere differentiable. The derivative does not exist at 0. So let’s consider the interval [imath](- \infty , \ 0).[/imath]

[math]f(x) = - |x|.\\ \text {But } - \infty < x < 0 \implies |x| = - x \implies f(x) = - (-x) = x \implies\\ f’(x) = 1, \ f(x) < 0, \text {and }\\ f(x) \text { is strictly increasing over } (- \infty, \ 0].[/math]
Now do the same for (0, 1). What is f(1)? Your conclusions?

How about (1, 2)? What is f(2)? How about if x > 2?
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I have to let the student think a little bit. ;)
 
Loki123 said:
View attachment 32510View attachment 32511
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The function is not strictly decreasing on all the intervals you circled.

This can't be determined merely from derivatives (which apply only to open intervals, not at points of discontinuity).

I believe that the only efficient way to determine the answer is to draw a graph of the piecewise function. (Any other method will be equivalent.) You may use derivatives as part of this work, but the important thing is to look at the discontinuities. Please try doing this.
 
Dr.Peterson said:
The function is not strictly decreasing on all the intervals you circled.

This can't be determined merely from derivatives (which apply only to open intervals, not at points of discontinuity).

I believe that the only efficient way to determine the answer is to draw a graph of the piecewise function. (Any other method will be equivalent.) You may use derivatives as part of this work, but the important thing is to look at the discontinuities. Please try doing this.
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i don't know how.
 
Loki123 said:
View attachment 32512
This is the best I can do.
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That's very close. But you show nothing between 2 and 3; and it isn't clear what's going on between 0 and 2. Also, I can't see any open circles, which will be important,

Did you read the page I referred to about graphing piecewise functions? Here's another:

Graph piecewise-defined functions | College Algebra

courses.lumenlearning.com courses.lumenlearning.com

Focus on example 13. This shows how to graph each piece separately, rather than looking only at end points.
 
@Loki123

You are GROSSLY underestimating yourself.

A function is a rule that generates a unique result for any given set of arguments.

A function that is defined over a real interval is a special kind of function. Is the function in this problem of that type?

A function that is continuous over some open interval is an even more special kind of function. Is the function in this problem of that type? Where?

A function that is differentiable over some open interval is a special kind of continuous function. Is the function in this problem of that type?

You can use properties of continuous functions wherever a function is continuous, but not elsewhere. You can use properties of differentiable functions wherever a continuous function is differentiable, but not elsewhere.

So you have intervals where what you know about continuous and differentiable functions apply. Use that knowledge.

There are four open intervals where this function is differentiable. What are they? Which derivatives, if any, indicate that the functions is strictly decreasing within those intervals?

That leaves three points that you must address directly through the virtually trivial definition of the function.

Given the definition of the function, which is defined for every real number, you now have all the information that you need to determine whether the function is strictly decreasing with various sets of argument.

My point is that you possess all the tools to address this question, but you are allowing its odd nature to distract from your ability to penetrate its oddity,
 
BigBeachBanana said:
@Loki123,

I know you can graph each function piece individually. Here’s a suggestion. Draw each piece as if there’s no restriction on its domain, then erase everything else leaving the interval behind. I’d recommend doing this with a pencil.
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My best shotIMG_20220502_234811.jpg
 
You're close enough that I'll make the needed corrections for you:

1651532585146.png​

Note the open dots showing points the graph is approaching. (If you're taking calculus, this should be a familiar idea.) You just sort of let the line trail off.

It would be a better idea to make it on graph paper, or at least make the grid evenly spaced.

Now use this to answer the question.
 
Dr.Peterson said:
You're close enough that I'll make the needed corrections for you:


Note the open dots showing points the graph is approaching. (If you're taking calculus, this should be a familiar idea.) You just sort of let the line trail off.

It would be a better idea to make it on graph paper, or at least make the grid evenly spaced.

Now use this to answer the question.
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But I don't get it. I didn't connect it to 2 because the domain for that line is x>2
 
Dr.Peterson said:
You're close enough that I'll make the needed corrections for you:


Note the open dots showing points the graph is approaching. (If you're taking calculus, this should be a familiar idea.) You just sort of let the line trail off.

It would be a better idea to make it on graph paper, or at least make the grid evenly spaced.

Now use this to answer the question.
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My answer stil stays the same. I don't know what this Changes.
 
Loki123 said:
But I don't get it. I didn't connect it to 2 because the domain for that line is x>2
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Please reread the link I gave you about how to graph piecewise functions. (Or study the topic in whatever textbook you have.) The open dot has a meaning. If that is not taught in your country, please show us an example of how you were taught to do this! (By which I mean an image of a graph from a textbook or other printed source, not just your attempt to copy it.)
Loki123 said:
My answer still stays the same. I don't know what this Changes.
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Please show your current answer, and explain it based on my correct graph (or your own, if you prefer). Then we can talk about whatever is wrong.
 
Dr.Peterson said:
Please reread the link I gave you about how to graph piecewise functions. (Or study the topic in whatever textbook you have.) The open dot has a meaning. If that is not taught in your country, please show us an example of how you were taught to do this! (By which I mean an image of a graph from a textbook or other printed source, not just your attempt to copy it.)

Please show your current answer, and explain it based on my correct graph (or your own, if you prefer). Then we can talk about whatever is wrong.
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I unfortunately do not have any material on this subject which is why I am so confused. I think I understand your graph better now that I thought about it, but my answer does not change.
the first interval given [1/3,1/2) i think is strictly decreasing because it's a subset of [0,1) where the graph is strictly decreasing.
the second interval given [0,1) is i think strictly decreasing as i have mentioned.
the third interval given [0,1] is not strictly decreasing in my opinion because it includes 1, and we can see what the line that includes 1 is not continuing the previous line but is starting from the top again.
the fourth interval [2, +infinity) is the same as the third. 2 belongs to a different line than +infinity which is why i don't think we can include it in the same interval as strictly decreasing.
the fifth interval (2,+infinity) does not have the problem as the fourth which is why i think it's strictly decreasing.
the sixth set {1,2,3,4,5,} is not strictly decreasing in my opinion because if i were to erase lines and only leave these points they would not be decreasing necessarily as 2 and 3 would be on the same horizontal line.
the seventh set {-1,4,5,6} is in my opinion strictly decreasing because if I were to erase lines and only leave those points they would be decreasing.
i realised i skipped over an interval, so i am just going to add it now.
the eight interval [1,+infitiny] is in my opinion not strictly decreasing because it decreases until 2 and then starts again at 2.
that's it. i have thought about this a lot and i do not see any other possibility.
 
Loki123 said:
I unfortunately do not have any material on this subject which is why I am so confused. I think I understand your graph better now that I thought about it, but my answer does not change.
the first interval given [1/3,1/2) i think is strictly decreasing because it's a subset of [0,1) where the graph is strictly decreasing.
the second interval given [0,1) is i think strictly decreasing as i have mentioned.
the third interval given [0,1] is not strictly decreasing in my opinion because it includes 1, and we can see what the line that includes 1 is not continuing the previous line but is starting from the top again.
the fourth interval [2, +infinity) is the same as the third. 2 belongs to a different line than +infinity which is why i don't think we can include it in the same interval as strictly decreasing.
the fifth interval (2,+infinity) does not have the problem as the fourth which is why i think it's strictly decreasing.
the sixth set {1,2,3,4,5,} is not strictly decreasing in my opinion because if i were to erase lines and only leave these points they would not be decreasing necessarily as 2 and 3 would be on the same horizontal line.
the seventh set {-1,4,5,6} is in my opinion strictly decreasing because if I were to erase lines and only leave those points they would be decreasing.
i realised i skipped over an interval, so i am just going to add it now.
the eight interval [1,+infitiny] is in my opinion not strictly decreasing because it decreases until 2 and then starts again at 2.
that's it. i have thought about this a lot and i do not see any other possibility.
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Yes, these are correct.
 
Loki123 said:
so then are my chosen intervals correct?
[1/3,1/2)
[0,1)
(2,+infinity)
{-1,4,5,6}
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Yes, when I say the answers and reasons are correct, that includes the answers.

In #32, I focused on the graph and didn't see the answers you showed there.
 
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