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!

### 1 Answer | Add Yours

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)

`[2x-4(x-1)]/x=(-2x+4)/x`

**Domain**=`(-oo,0)U(0,+oo)`

`f(f(x))=f(x/(x-1))=[x/(x-1)]/[x/(x-1)-1]=` (Multiply numerator and denominator by x-1)

`x/[x-(x-1)]=x/1=x`

**Domain**=`(-oo,+oo)`

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