Another discussion about notation

pka

pka

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Lynxbci said:
Given that f(x) = 3x+1 and g(x) = x^2
solve fg(x) = gf(x)
Fg(x) 3x^2+1
gf(x) 9x^2 +6x+1
but cannot solve, help please?
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In my experience in courses dealing with functions the notation \(\displaystyle fg(x)\) means \(\displaystyle f(x)\cdot g(x)\).
Just as \(\displaystyle f+g(x)\) means \(\displaystyle f(x)+g(x)\) or the addition of functions then \(\displaystyle fg(x)\) means multiplication of functions.
In this case \(\displaystyle fg(x)=gf(x)=3x^3+x^2\).

It appears that you are assuming that it is \(\displaystyle f\circ g(x)=f(g(x))=3(x^2)+1\).

So which is it, multiplication or composition?
 
pka said:
In my experience in courses dealing with functions the notation \(\displaystyle fg(x)\) means \(\displaystyle f(x)\cdot g(x)\).
Just as \(\displaystyle f+g(x)\) means \(\displaystyle f(x)+g(x)\) or the addition of functions then \(\displaystyle fg(x)\) means multiplication of functions.
In this case \(\displaystyle fg(x)=gf(x)=3x^3+x^2\).

It appears that you are assuming that it is \(\displaystyle f\circ g(x)=f(g(x))=3(x^2)+1\).

So which is it, multiplication or composition?
Click to expand...

Since multiplication is commutative, it would be a silly problem if we interpret it as multiplication of functions.

I searched for "solve fg(x) = gf(x)" and found several sources, including a British textbook, that clearly define fg as composition, so I think we can safely assume that interpretation. It also yields a simple solution.

Lynxbci, have you worked it out based on Subhotosh Khan's help?
 
Dr.Peterson said:
Since multiplication is commutative, it would be a silly problem if we interpret it as multiplication of functions
I searched for "solve fg(x) = gf(x)" and found several sources, including a British textbook, that clearly define fg as composition, so I think we can safely assume that interpretation. It also yields a simple solution.
Lynxbci, have you worked it out based on Subhotosh Khan's help?
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SEE HERE. I will admit that elementary textbooks do use parentheses \(\displaystyle (f\cdot g)(x) \) for multiplication.
But I think it important to point out ambiguities in notation usage.
BTW I have taught in the UK, I never saw that notation used.
 
pka said:
SEE HERE. I will admit that elementary textbooks do use parentheses \(\displaystyle (f\cdot g)(x) \) for multiplication.
But I think it important to point out ambiguities in notation usage.
BTW I have taught in the UK, I never saw that notation used.
Click to expand...

I haven't seen it used at an elementary level either, though Wikipedia mentions it here.

One place I found it is here (page 7, section 2.3). And problem 10 here is the same type.

And, yes, the variability of notation makes it important to ask what someone means. It can vary not only country to country, but author to author.
 
mmm4444bot said:
If it's important to ask, why is it safe to assume?
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Because, as I stated, (a) it is a known option; (b) the other option would not make a reasonable question; and (c) the asker clearly thinks that is the meaning of the notation, knowing the context as we don't.

But I did say "think", not "know".
 
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