For the functions Find the composite fog and simplify

Question

f(x) = x/(x+3 and g(x) = 5/(x-1)

find(fog)(x) and domain of (fog)(x)

Answer 1

Given functios are f(x) = x/(x+3) and g(x) = 5/(x-1)

fog(x)  =  f( g(x) )

             =  f( 5/(x-1) )                                                      [ Since g(x) = 5/(x-1) ]

             =  ( 5/(x-1) ) / [( 5/(x-1) ) + 3]                             [ Since f(x) = x/(x+3) ]

             =  ( 5/(x-1) ) / [( 5 + 3(x - 1))/(x-1) )]   

             =  5 / [( 5 + 3(x - 1)]

             =  5 / ( 5 + 3x - 3)

fog(x)   =  5 / (3x + 2)

Domain is all the possible values of x

In the compsite function if denominator is zeor it is undefined,

Hence, Equate the denominator to zero and find out the value at which it is zero.

Then exclude that value from domain.

3x + 2  =  0

3x  =  -2

x  =  -2/3

At x = -2/3 the the denominator is zero.

Therefore, Domain : (-infinity, -2/3)  U  ( -2/3, infinity)

Answer : 

fog(x)   =  5 / (3x + 2)

Domain : (-infinity, -2/3)  U  ( -2/3, infinity)