Help with Understanding Composite Functions

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    Composite Functions
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Composite functions involve taking the output of one function and using it as the input for another, specifically expressed as f(g(x)). The confusion often arises between composite functions and multiplication of functions; they are distinct operations. For example, if g(x) = x + x^2 and f(x) = 2x, then the composition f(g(x)) results in f(g(x)) = 2(x + x^2). Understanding that the output of g(x) becomes the input for f is crucial for correctly determining the composite function. Clarifying these concepts helps in accurately finding the domain and evaluating composite functions.
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Homework Statement



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The Attempt at a Solution


f+g would be

(2x^2+1) + (x-1) = 2x^2 + x so the domain for f+g is all real numbers but i don't know how to find the one for the composite. i am still confused as to what a composite function is, please help me! thank you!
 
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composition of f&g is nothing but f o g which is the same as fg or f(g(x)).

So take the entire g(x) and put it wherever you see 'x' in f.
 


the composite is simply g then? or is it f(x) multiplied by g(x)?
 


frozenbananas said:
the composite is simply g then? or is it f(x) multiplied by g(x)?

As rock.freak said, it is f(g(x)), not g(x). 'x' is the input to g and then the output g(x) becomes the input to f. It is different to multiplying.

As an example: If g(x)=x+x^2 and f(x)=2x, then the composition gf(x) = 2x + (2x)^2
 

Thread 'Finding the number of ways to arrange identical balls in a circle (3 different colors)'

I tried to combine those 2 formulas but it didn't work. I tried using another case where there are 2 red balls and 2 blue balls only so when combining the formula I got ##\frac{(4-1)!}{2!2!}=\frac{3}{2}## which does not make sense. Is there any formula to calculate cyclic permutation of identical objects or I have to do it by listing all the possibilities? Thanks
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Thread 'Greatest possible value of a constant in polynomial'

Since ##px^9+q## is the factor, then ##x^9=\frac{-q}{p}## will be one of the roots. Let ##f(x)=27x^{18}+bx^9+70##, then: $$27\left(\frac{-q}{p}\right)^2+b\left(\frac{-q}{p}\right)+70=0$$ $$b=27 \frac{q}{p}+70 \frac{p}{q}$$ $$b=\frac{27q^2+70p^2}{pq}$$ From this expression, it looks like there is no greatest value of ##b## because increasing the value of ##p## and ##q## will also increase the value of ##b##. How to find the greatest value of ##b##? Thanks
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