How do you find the compositions given $f(x) = x/(x^2+1)$ , $g (x) = x^2 +1$ ?

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Answer 1 · 2016-09-15T16:26:24

$f(g(x))=(x^2+1)/(x^4+2x^2+2)$

$g(f(x))=(x^4+3x^2+1)/(x^4+2x^2+1)$

The compositions are $f(g(x))$ and $g(f(x))$

Let's find $f(g(x))$
$f(g(x))=(x^2+1)/((x^2+1)^2+1)$

$=(x^2+1)/(x^4+2x^2+1+1)$

$=(x^2+1)/(x^4+2x^2+2)$

Let's find $g(f(x))$
$g(f(x))=(x/(x^2+1))^2+1$
$g(f(x))=(x^2/(x^4+2x^2+1))+1$

$g(f(x))=(x^2+x^4+2x^2+1)/(x^4+2x^2+1)$

$g(f(x))=(x^4+3x^2+1)/(x^4+2x^2+1)$

How do you find the compositions given f(x) = x/(x^2+1)f(x) = x/(x^2+1), g (x) = x^2 +1g (x) = x^2 +1?