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

Answer 1

Elias Ackerman

For f(x) the domain is #R-{1}# and for g(x) is #R# where #R# is the set of real numbers. Hence

#f(g(x))=(x^2-1)/(x^2-1+1)=(x^2-1)/(x^2)=1-1/(x^2)#

and

#g(f(x))=(x/(x+1))^2-1#

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Answer 2

Chloe Alexander

To find the composition of functions ( f(x) ) and ( g(x) ), denoted as ( f \circ g ) and ( g \circ f ), follow these steps:

For ( f \circ g ), substitute ( g(x) ) into ( f(x) ), meaning replace every occurrence of ( x ) in ( f(x) ) with ( g(x) ).
For ( g \circ f ), substitute ( f(x) ) into ( g(x) ), meaning replace every occurrence of ( x ) in ( g(x) ) with ( f(x) ).

Given ( f(x) = \frac{x}{x + 1} ) and ( g(x) = x^2 - 1 ), we can find the compositions as follows:

( f \circ g ): Substitute ( g(x) = x^2 - 1 ) into ( f(x) ): [ f \circ g = f(g(x)) = f(x^2 - 1) = \frac{x^2 - 1}{x^2 - 1 + 1} = \frac{x^2 - 1}{x^2} ]
( g \circ f ): Substitute ( f(x) = \frac{x}{x + 1} ) into ( g(x) ): [ g \circ f = g(f(x)) = g\left(\frac{x}{x + 1}\right) = \left(\frac{x}{x + 1}\right)^2 - 1 = \frac{x^2}{(x + 1)^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.