Given: f(x)= x/(x-1) and g(x)=(2x-4)/x find the following and their domains: f(g(x)) , g(f(x)) , f(f(x))Can you please explain the steps, thank you!
In the following work we will be dealing with complex fractions, I will be simplifying the problem by multiplying by the LCD.
`f(g(x))=f([2x-4]/x)=[(2x-4)/x]/[(2x-4)/x-1]=` (Multiply the top and bottom by x)
`[2x-4]/[(2x-4)-x]=[2x-4]/[x-4]` This function is defined as long as x is not 4.
Thus domain is `(-oo,4)U(4,+oo)`
`g(f(x))=g(x/(x-1))=[2(x/(x-1))-4]/(x/(x-1))=` (Multiply top and bottom by x-1)
`f(f(x))=f(x/(x-1))=[x/(x-1)]/[x/(x-1)-1]=` (Multiply numerator and denominator by x-1)