Finding f/g: Composite Functions

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    Composite Functions
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Discussion Overview

The discussion revolves around finding the composite function \( f \circ g \) and the quotient \( \frac{f}{g} \) for the functions \( f(x) = x^2 + 1 \) and \( g(x) = \frac{1}{x} \). Participants explore the calculations and expressions related to these functions.

Discussion Character

  • Mathematical reasoning
  • Technical explanation

Main Points Raised

  • One participant expresses difficulty in understanding how to find \( \frac{f}{g} \) despite knowing the answer.
  • Another participant provides a solution for the quotient, stating it simplifies to \( (x^2 + 1)x = x^3 + x \).
  • One participant correctly computes the composite function as \( f(g(x)) = \frac{1}{x^2} + 1 \).
  • A later reply reiterates the calculation of the composite function, showing the intermediary step of substituting \( g(x) \) into \( f(x) \).

Areas of Agreement / Disagreement

Participants present multiple approaches to the problem, with no consensus on the primary focus of the discussion, as both the composite function and the quotient are explored.

Contextual Notes

The discussion includes various expressions and calculations that may depend on the interpretation of composite functions versus quotients, which could lead to confusion regarding the intended question.

needhelpplease
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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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