Function notation

[jpanknin]

Can someone please explain the [imath]f^2(c)[/imath] notation in the screenshot below? I've not seen this notation and according to Google searches it could be either a function iterate or squaring the output. The context is the continuity of a function, but I'm only asking about the notation.

 

[fresh_42]

It normally means [imath] f^2(c)=f(f(c)). [/imath] I would write [imath] (f(c))^2=f(c)^2, [/imath] but this may depend on the author, where he puts the square. E.g., if [imath] f(x)=\sin(x) [/imath] then everybody reads [imath] \sin^2(x) [/imath] as [imath] (\sin(x))^2 [/imath] and not as applying the sine function twice. This is true for all standard functions. If we only have a general [imath] f [/imath] then positioning the square without additional parentheses is ambiguous. You can normally see it from context.

 

[jpanknin]

It normally means [imath] f^2(c)=f(f(c)). [/imath] I would write [imath] (f(c))^2=f(c)^2, [/imath] but this may depend on the author, where he puts the square. E.g., if [imath] f(x)=\sin(x) [/imath] then everybody reads [imath] \sin^2(x) [/imath] as [imath] (\sin(x))^2 [/imath] and not as applying the sine function twice. This is true for all standard functions. If we only have a general [imath] f [/imath] then positioning the square without additional parentheses is ambiguous. You can normally see it from context.

Yes, I normally see the square written as [imath](f(x))^2[/imath] or similar. I've gone through the book and don't see any similar notation. This question is on a separately developed homework though, so likely a different author.

 

[fresh_42]

It depends a bit on whether [imath] f^2(c)=(f\circ f)(c)=f(f(c)) [/imath] is an option at all. The specific line looks more as if [imath] f^2(c)=f(c)^2 [/imath] was meant. Applying a sine on a sine usually doesn't make sense. It is more problematic if you deal with complexity theory, where time and space of algorithms are calculated. They have a lot of [imath] \log(\log(n)) [/imath] expressions. But I think they are normally written out and [imath] \log^2(n) [/imath] means indeed [imath] (\log(n))^2 .[/imath]

[imath] f^2(c)= (f\circ f)(c)=f(f(c))[/imath] is rare. It occurs when you investigate function spaces (algebras) or sometimes in functional identities. If I were to bet, I would put my money on [imath] f^2(c)=1/4=(f(c))^2, [/imath] but it is not ultimately clear.

 

[jpanknin]

It depends a bit on whether [imath] f^2(c)=(f\circ f)(c)=f(f(c)) [/imath] is an option at all. The specific line looks more as if [imath] f^2(c)=f(c)^2 [/imath] was meant. Applying a sine on a sine usually doesn't make sense. It is more problematic if you deal with complexity theory, where time and space of algorithms are calculated. They have a lot of [imath] \log(\log(n)) [/imath] expressions. But I think they are normally written out and [imath] \log^2(n) [/imath] means indeed [imath] (\log(n))^2 .[/imath]

[imath] f^2(c)= (f\circ f)(c)=f(f(c))[/imath] is rare. It occurs when you investigate function spaces (algebras) or sometimes in functional identities. If I were to bet, I would put my money on [imath] f^2(c)=1/4=(f(c))^2, [/imath] but it is not ultimately clear.

Ok, appreciate the replies and help.

 

[Dr.Peterson]

Can someone please explain the [imath]f^2(c)[/imath] notation in the screenshot below? I've not seen this notation and according to Google searches it could be either a function iterate or squaring the output. The context is the continuity of a function, but I'm only asking about the notation.

View attachment 39939

Please show the actual context! Since the meaning of the notation depends on that, we really can't give a definite answer without that. So far it's just a guess.

So, what instructions are given for this problem or group of problems? What do the other problems look like? Actually, what does the rest of this problem say??

I did a search to see if the whole problem happened to be available, and I found it:


It is still not entirely obvious which meaning is appropriate, except that we don't know enough to know what [imath]f(f(-2))[/imath] is, so it appears to mean [imath](f(x))^2[/imath]. (And whoever gave the answer I found does take it that way.)

But I can't imagine why anyone would write this problem in either case, especially without ever having used the notation in the context!

 

[jpanknin]

Please show the actual context! Since the meaning of the notation depends on that, we really can't give a definite answer without that. So far it's just a guess.

So, what instructions are given for this problem or group of problems? What do the other problems look like? Actually, what does the rest of this problem say??

I did a search to see if the whole problem happened to be available, and I found it:

View attachment 39940
It is still not entirely obvious which meaning is appropriate, except that we don't know enough to know what [imath]f(f(-2))[/imath] is, so it appears to mean [imath](f(x))^2[/imath]. (And whoever gave the answer I found does take it that way.)

But I can't imagine why anyone would write this problem in either case, especially without ever having used the notation in the context!

This is a homework problem, so I hesitated to put the whole thing up. But yes, that's the question (slightly different answer choices, but very similar). I answered under the assumption that it meant [imath](f(x))^2[/imath] and it was correct. Glad I'm not the only one who was confused! Thanks very much for the responses.