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)#