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

Answer 1

To find the composition ( (f \circ g)(x) ), substitute ( g(x) ) into ( f(x) ). Thus, ( (f \circ g)(x) = f(g(x)) = f(x^2 + 1) ). Therefore, ( (f \circ g)(x) = \frac{x^2 + 1}{(x^2 + 1)^2 + 1} ).

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Answer 2

Elijah Allen

#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)#

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.