Question

HELP ASAP PLZ i need help !
help me :((

Answer 1

(1)
`g[f(x)]=(8x-1)/(12x-4)`

(2)
`g^(-1)(x)=(x+1)/((3x-2))`

(3)
`x=(9+√(129) )/(6), (9-√(129) )/(6)`

Explanation:

Given functions f(x) = (4x-1)

are
`g(x)=(2x+1)/(3x-1)`

(1)
`g[f(x)]=(2(4x-1)+1)/(3(4x-1)-1)`

`=(8x-2+1)/(12x-3-1)`

`=(8x-1)/(12x-4)`

(2) for
`g^(-1)(x)`, rewrite the function g(x) in terms of an equation

`y=(2x+1)/(3x-1)`

Substitute y in place of x and x in place of y, then solve for y.

`x=(2y+1)/(3y-1)`

(3y-1)x = 2y+1

3xy - x = 2y + 1

3xy - 2y = x + 1

y(3x-2) = x + 1

`y=((x+1)/(3x-2) )`

⇒
`g^(-1)(x)=(x+1)/((3x-2))`

(3)
`f[g(x)]=(x-1)`

`f[g(x)]=4[(2x+1)/(3x-1)] =(x-1)`

⇒
`((8x+4)-(3x-1))/(3x-1)=(x-1)`

⇒
`(8x-3x+4+1)/(3x-1)=(x-1)`

⇒
`(5x+5)/((3x-1))=(x-1)`

⇒
`5x+5=(3x-1)(x-1)`

⇒
`5x+5=3x^2-3x-x+1`

⇒
`5x+5=3x^2-4x+1`

⇒
`3x^2-9x-4=0`

⇒
`x=(9\pm√((-9)^2-4(-4)*3) )/(2*3)`

`x=(9\pm√(81+48) )/(6)`

`x=(9\pm√(129) )/(6)`

`x=(9+√(129) )/(6), (9-√(129) )/(6)`