How do you find f(g(x)) if #g(x) = 3/(x - 1)# and #f(x) = (x - 1)/(x - 3)#?

Answer 1

To find ( f(g(x)) ) given that ( g(x) = \frac{3}{x - 1} ) and ( f(x) = \frac{x - 1}{x - 3} ), we substitute the expression for ( g(x) ) into ( f(x) ) wherever we see ( x ) in ( f(x) ).

So, we have:

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

Now, substitute ( g(x) ) into ( f(x) ):

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

[ = \frac{\frac{3 - (x - 1)(x - 1)}{x - 1}}{\frac{3 - 3(x - 1)}{x - 1}} ]

[ = \frac{3 - x + 1}{3 - 3x + 3} ]

[ = \frac{4 - x}{3 - 3x + 3} ]

[ = \frac{4 - x}{6 - 3x} ]

So, ( f(g(x)) = \frac{4 - x}{6 - 3x} ).

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Answer 2

Abigail Allen

The answer is #=(4-x)/(6-3x)#

This is a composition of functions

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

#g(x)=3/(x-1)#

#f(g(x))=f(3/(x-1))#

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

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

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

#=(4-x)/(6-3x)#

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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.