Question

Question #43f3b

Answer 1

`x=1/10`

Explanation:

f(g(x)=`4 (6x)/(x-1) +2= (24x)/(x-1) +2`

Given, f(g(x) = g(x), hence,

`(24x)/(x-1) +2= (6x)/(x-1)`

`(18x)/(x-1) +2=0`

18x +2(x-1)=0

20x -2=0

20x=2

`x=1/10`

Answer 2

Given: `f(x) = 4x+2` and `g(x)= 6x/(x-1)`

`f(g(x)) = 4g(x)+2`

Normally, we would substitute `6x/(x-1)` for `g(x)` and then simplify but let's wait a bit to do that.

Instead, let's set `4g(x)+2` equal to `g(x)`:

`4g(x)+2=g(x)`

Subtract `g(x)+2` from both sides:

`3g(x)= -2`

Now let's substitute `6x/(x-1)` for `g(x)`:

`3(6x/(x-1))= -2`

Multiply both sides by `(x-1)`:

`3(6x)= -2(x-1)`

Distribute through the ()s:

`18x= -2x+2`

Add 2x to both sides:

`20x= 2`

Divide both sides by 20:

`x= 1/10`

Check

`g(1/10) = 6(1/10)/(1/10-1)`

`g(1/10) = -2/3`

`f(-2/3) = 4(-2/3)+ 2= -2/3`

`4(-2/3)+2 = -2/3`

`-8/3+6/2 = -2/3`

`-2/3 = -2/3`

This checks.

Answer 3

`x=1/10.`

Explanation:

`f(x)=4x+2 rArr f(g(x))=4g(x)+2.`

But, given that, `f(g(x))=g(x) rArr 4g(x)+2=g(x) rArr 3g(x)=-2`

`because, g(x)=(6x)/(x-1), :. 3{(6x)/(x-1)}=-2`

`rArr 18x=-2(x-1)=2-2x rArr 18x+2x=2 rArr 20x=2 rArr x=2/20=1/10.`