Let #f(x) = 1 /(x+3)# and #g(x) = -2 /x#, how do you find each of the compositions?

Answer 1

To find the compositions ( f \circ g ) and ( g \circ f ), you substitute the expression of one function into the other and simplify:

( f \circ g ): [ (f \circ g)(x) = f(g(x)) = f\left(-\frac{2}{x}\right) = \frac{1}{-\frac{2}{x} + 3} ]
( g \circ f ): [ (g \circ f)(x) = g(f(x)) = g\left(\frac{1}{x + 3}\right) = -\frac{2}{\frac{1}{x + 3}} ]

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Answer 2

Anna Allen

Please see the explanation below.

The functions are

#f(x)=1/(x+3)#

And

#g(x)=-2/x#

There are #2# compositions

#fog(x)=f(g(x))=f(-2/x)#

#=1/((-2/x)+3)#

#=x/(3x-2)#

And

#gof(x)=g(f(x))=g(1/(x+3))#

#=-2/(1/(x+3))#

#=-2(x+3)#

Therefore,

#fog(x)!=gof(x)#

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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.