# Exponential Growth at Most

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I was just wondering can I use the 'if then' statement in a way that it gives me a rule to check for how a function is not exponential growth at most γ? Or for that would I need the converse to be true and then find its negation?

Also, does the contrapositive tell me a case 'when f is not of exponential growth at most γ'? Like it doesn't tell me how to find it, but it would tell me that if 'f is not of exponential growth'. then it must be that ... the negation of 'f is continuous ... and ...' holds. So, if it didn't hold, we know it must be of exponential growth?

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I was just wondering can I use the 'if then' statement in a way that it gives me a rule to check for how a function is not exponential growth at most γ? Or for that would I need the converse to be true and then find its negation?

**Takeover Season**)I was just wondering can I use the 'if then' statement in a way that it gives me a rule to check for how a function is not exponential growth at most γ? Or for that would I need the converse to be true and then find its negation?

You currently have , but what you seem to be after is the form .

The only way this can be obtained is if we had

because the equivalent contrapositive of this is just .

Although, it is useful to note that what you're given is a definition of the new terminology:

" is of expenential growth at most ."

And every definition is an 'if and only if' statement, meaning converses hold.

Hence, it is perfectly valid in your case to say that

.

All this really says, in very simple terms, is that if the conditions of the definition aren't satisfied, you can't use that phrase. Very simple logic.

Also, does the contrapositive tell me a case 'when f is not of exponential growth at most γ'? Like it doesn't tell me how to find it, but it would tell me that if 'f is not of exponential growth'. then it must be that ... the negation of 'f is continuous ... and ...' holds. So, if it didn't hold, we know it must be of exponential growth?

Contrapositive tells you that if is not of exponential growth at most , then either is discontinuous OR is unbounded.

In a more symbolic sense;

The contrapositive of

is

Last edited by RDKGames; 6 months ago

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(Original post by

You need converse to be true.

You currently have , but what you seem to be after is the form .

The only way this can be obtained is if we had

because the equivalent contrapositive of this is just .

Although, it is useful to note that what you're given is a definition of the new terminology:

" is of expenential growth at most ."

And every definition is an 'if and only if' statement, meaning converses hold.

Hence, it is perfectly valid in your case to say that

.

All this really says, in very simple terms, is that if the conditions of the definition aren't satisfied, you can't use that phrase. Very simple logic.

Yes.

Contrapositive tells you that if is not of exponential growth at most , then either is discontinuous OR is unbounded.

In a more symbolic sense;

The contrapositive of

is

**RDKGames**)You need converse to be true.

You currently have , but what you seem to be after is the form .

The only way this can be obtained is if we had

because the equivalent contrapositive of this is just .

Although, it is useful to note that what you're given is a definition of the new terminology:

" is of expenential growth at most ."

And every definition is an 'if and only if' statement, meaning converses hold.

Hence, it is perfectly valid in your case to say that

.

All this really says, in very simple terms, is that if the conditions of the definition aren't satisfied, you can't use that phrase. Very simple logic.

Yes.

Contrapositive tells you that if is not of exponential growth at most , then either is discontinuous OR is unbounded.

In a more symbolic sense;

The contrapositive of

is

**OR**, then we have that is not of exponential growth at most . So, we need one of the two of P and Q negations to hold to show that is not of exponential growth at most ?

Also, for the last part, if is discontinuous, then does this mean over the interval, that

**f**is not continuous at, at least, 1 point in ?

Now, so looking at this "Contrapositive tells you that if is not of exponential growth at most , then either is discontinuous OR is unbounded.".

Can I say that, checking that either is discontinuous OR is unbounded and so it must be that is not of exponential growth at most . Or is this flipping the implication around again? I sort of know that if is not of exponential growth at most , then one of those negations must hold. So, by showing 1 of them hold, is it okay to conclude is not of exponential growth at most ?

Last edited by Takeover Season; 6 months ago

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So, we need one of the two of P and Q negations to hold to show that is not of exponential growth at most ? Also, for the last part, if is discontinuous, then does this mean over the interval, that

**Takeover Season**)So, we need one of the two of P and Q negations to hold to show that is not of exponential growth at most ? Also, for the last part, if is discontinuous, then does this mean over the interval, that

**f**is not continuous at, at least, 1 point in ?
Now, so looking at this "Contrapositive tells you that if is not of exponential growth at most , then either is discontinuous OR is unbounded.".

Can I say that, checking that either is discontinuous OR is unbounded and so it must be that is not of exponential growth at most . Or is this flipping the implication around again?

Can I say that, checking that either is discontinuous OR is unbounded and so it must be that is not of exponential growth at most . Or is this flipping the implication around again?

Sounds to me like you're paying too much attention into the logical specifics of an 'if and only if' statement which is just a basic definition. Not really sure why you care.

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(Original post by

Yes, don't overthink it.

Yes it flips. What you said is the converse of the contrapositive.

Sounds to me like you're paying too much attention into the logical specifics of an 'if and only if' statement which is just a basic definition. Not really sure why you care.

**RDKGames**)Yes, don't overthink it.

Yes it flips. What you said is the converse of the contrapositive.

Sounds to me like you're paying too much attention into the logical specifics of an 'if and only if' statement which is just a basic definition. Not really sure why you care.

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