MHB Finding f/g: Composite Functions

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To find the composite function f(g(x)), where f(x)=x^2+1 and g(x)=1/x, the calculation results in f(g(x))=f(1/x)=(1/x)^2+1, which simplifies to 1/x^2+1. Additionally, the quotient of the functions is calculated as f/g(x)=(f(x))/(g(x))=(x^2+1)/(1/x), leading to the expression (x^2+1)x=x^3+x. The discussion clarifies the distinction between composite functions and the quotient of functions. The final results for both operations are provided clearly.
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The questions is asking me to find \frac{f}{g} basically , the question is asking me to find the answer , even though i know it, i can't get my head around it.

the composite function is

f(x)=x^2+1
g(x)=1/x

we need to find foG (f of g) [composite functions].
 
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HelpPlease said:
The questions is asking me to find \frac{f}{g} basically , the question is asking me to find the answer , even though i know it, i can't get my head around it.

the composite function is

f(x)=x^2+1
g(x)=1/x

we need to find foG (f of g) [composite functions].

Thank you solved.Divide them both so x^2+1 / 1/x
switch them to multiply so it's going to be x^2+1/x
 
$f \circ g(x)= f(g(x))=\frac{1}{x^2}+1$
 
Additionally, the quotient (which is not the composite $f \circ g$) is:
$$\frac fg(x) = \frac{f(x)}{g(x)}=\frac{x^2+1}{1/x}=(x^2+1)x=x^3+x$$
 
Just to add an intermediary step:

$$(f\circ g)(x)=f(g(x))=f\left(\frac{1}{x}\right)=\left(\frac{1}{x}\right)^2+1=\frac{1}{x^2}+1$$
 
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