anonymous
  • anonymous
Find gof if f(x)=x+(1/x) and g(x)= (x+1)/(x+2)
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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jamiebookeater
  • jamiebookeater
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princeharryyy
  • princeharryyy
g(f(x))= (f(x)+1)/(f(x)+2)
princeharryyy
  • princeharryyy
just out the value of f(x) in place of f(x) in above equation n solve
princeharryyy
  • princeharryyy
@Theanitrix

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princeharryyy
  • princeharryyy
put*
anonymous
  • anonymous
I know that, I just am not sure how to solve it
princeharryyy
  • princeharryyy
ok
anonymous
  • anonymous
the fraction confuses me
Jhannybean
  • Jhannybean
\[(g \circ f) = g\left(x+\frac{1}{x}\right) = \frac{\left(x+\dfrac{1}{x}\right)+1}{\left(x+\dfrac{1}{x}\right)+2}\]
princeharryyy
  • princeharryyy
well, you got the answer now, I guess.
Jhannybean
  • Jhannybean
That isn't the answer.
princeharryyy
  • princeharryyy
that's almost the final answer @Jhannybean
anonymous
  • anonymous
the answer would be (x^2+x+1)/(x+1)^2
anonymous
  • anonymous
but i want to know how they got that answer
Jhannybean
  • Jhannybean
multiply both the numerator and denominator by \(x\). \[\frac{\left(x+\dfrac{1}{x}\right)+1}{\left(x+\dfrac{1}{x}\right)+2} \cdot \frac{x}{x} \]
princeharryyy
  • princeharryyy
(g∘f)=g(x+1/x)=(x^2+1+x)/(x^2+2+x)
anonymous
  • anonymous
is that because it is the denominator of 1/x? @Jhannybean
Jhannybean
  • Jhannybean
Yes @Theanitrix
anonymous
  • anonymous
okay so would that same thing apply to fof?
Jhannybean
  • Jhannybean
\[(f\circ f)= f\left(x+\frac{1}{x}\right) = \left(x+\frac{1}{x}\right)+\dfrac{1}{x+\dfrac{1}{x}}\]
Jhannybean
  • Jhannybean
but when you're trying to solve this... set all that complicated junk = \(a\) or any other variable. find the common denominator, and simplify.
Jhannybean
  • Jhannybean
\[\text{let a}=x+\frac{1}{x}\]\[(f\circ f) = a+\frac{1}{a} = \frac{a^2+1}{a} = \dfrac{\left(x+\dfrac{1}{x}\right)^2+1}{\left(x+\dfrac{1}{x}\right)} \]
Jhannybean
  • Jhannybean
\[=\frac{x^2+\dfrac{1}{x^2}+2 +1}{\dfrac{x^2+1}{x}} =\frac{x^2+\dfrac{1}{x^2}+3}{\dfrac{x^2+1}{x}} \] So now you'd want to multiply both numerator and denominator by \(\dfrac{x}{x}\)
anonymous
  • anonymous
i did it a different way
anonymous
  • anonymous
and got the answer x^4+3x^2+1/x(x^2+1)
Jhannybean
  • Jhannybean
Thats correct
anonymous
  • anonymous
okay, i understood your approach as well so thank you
Jhannybean
  • Jhannybean
\[\frac{x^2+\dfrac{1}{x^2}+3}{\dfrac{x^2+1}{x}} \cdot \frac{x}{x} =\frac{\dfrac{x^4+3x^2+1}{x}}{x^2+1} = \color{red}{\frac{x^4+3x^2+1}{x(x^2+1)}}\]

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