Question

Find the inverse of the following function:


`f(x)=-2 √(3x-1) -5`
if x \geq 1/3[/tex]

Answer 1

First, we are going to find the inverse function of
`f(x)` by replacing
`f(x)` with
`y`, and then, interchanging
`x` and
`y` and solving for
`x`:

`y=-2 √(3x-1) -5`

`x=-2 √(3y-1) -5`

`-2 √(3y-1) =x+5`

`√(3y-1) = (x+5)/(-2)`

`3y-1=( (x+5)/(-2) )^(2)`

`3y=( (x+5)/(-2) )^(2) +1`

`y= (( (x+5)/(-2))^(2)+1 )/(3)`

Next we are going to evaluate our inverse function at
`x= (1)/(3)`:

`y= (( ( (1)/(3)+5 )/(-2))^(2)+1 )/(3)`

`y= (( ( (16)/(3) )/(-2))^(2)+1 )/(3)`

`y= ((- (8)/(3))^(2)+1 )/(3)`

`y= ( (64)/(9)+1 )/(3)`

`y= ( (73)/(9) )/(3)`

`y= (73)/(27)`

We can conclude that the inverse function of
`f(x)=-2 √(3x-1) -5` when
`x= (1)/(3)` is
`(73)/(27)`