#How to demonstrate a property?

40 messages · Page 1 of 1 (latest)

fervent vigil
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He asked me to verify if function f is bijective or not
Then to draw it and its reciprocal function
Well at the end without calculating the reciprocal function I must demonstrate that property
In the correction they used the derivative and a whole another function
I have no idea why they chose that approach
So I want an explanation or another approach, I'm all ears for all ideas
Thank you in advance for any help

rn_image_picker_lib_temp_0e9583fa-3a8e-431d-9ab9-c440f7de72e6.jpg rn_image_picker_lib_temp_dd70a3d1-9d6e-40d5-b558-25d5ff43d7db.jpg
lyric ginkgoBOT
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still whale
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Why are your pictures upside down?

fervent vigil
still whale
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,rotate

fervent vigil
still whale
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A function f is the inverse of a function g if and only if f(g(x)) = x.

fervent vigil
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Yes this is the one I'm talking about
Thanks for the clarification
(I guess the notation/ names are different from french to English, my bad for the wrong name)

still whale
fervent vigil
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In french: " l'inverse d'une fonction " is 1/f
" La fonction reciproque " is the inverse function

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The similarities between the two words in french and English got me confused
And I apologize for my lack of knowledge about those mathematics terms in English

still whale
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That's really weird, because 1/f is the "reciprocal" of f in English. That's literally what the word "reciprocal" means.

fervent vigil
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Well that's one fascinating thing about languages I guess...
Anyways back to the original question, putting the language differences aside,
Why did he derive the sum of those two inverse functions... I didn't quite grasp the approach he used to answer the last question
It would be kind if you if you can explain it or suggest another approach (without calculating the inverse function or knowing it's equation, that's the whole point of the question... At this level we're limited only to some of the inverse functions not all of them, so using cos-¹ or sin-¹...etc is not allowed at this level yet)
Thanks for your patience and help

still whale
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"Sum of those two inverse functions"?

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,rotate

young abyssBOT
still whale
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So I can't read the French. Could you translate?

still whale
arctic idol
still whale
arctic idol
fervent vigil
# still whale So I can't read the French. Could you translate?

We got f defined on [0,1] and f(x) cos(πx)
Demonstrate that f is bijective from [0,1] on an interval J we are going to precise
Represent the two function, f and inverse of f in the same Orthonormal reference
Demonstrate that for every x in J we got
f-¹(x) + f-¹(-x) = 1
What can we deduce graphically

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In the correction
They created another function, let's say g,
g = f-¹(x) + f-¹ (-x)

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They showed that this function is Derivable and bijective on [0,1]
Calculated the derivative, and showed that it always equal to 0
So g is a constant
Then they calculated g(1) and found out it equal to 1 so I am understanding this approach properly and that's why I'm asking why did they do this specially? And if there another approach?

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My apologies for disappearing all of a sudden yesterday and thanks again for your patience

still whale
fervent vigil
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Creating this new function (the sum) and then deriving it
What did they have in mind?
Surely they used this method for its specificity, or is it a random solution they found after trying many times and I have to apply it without thinking everytime I find this type of question?
I wanted to know the reasoning behind it if possible

still whale
fervent vigil
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Ohh so that's it what he had in mind
Well this might be helpful in the future

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Thank you for your help and patience

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I want to thank you with the bot but I don't know how
Do I just mention him?
And then "+close"?

still whale
fervent vigil
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Okay thank you once again

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+close

robust treeBOT
# fervent vigil +close
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# robust tree

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