Question

Find the inverse function of f(x) = x+2/x+1

Answer 1

`f^(-1)`(x) =
`(2-x)/(x-1)`

Explanation:

Let y = f(x) and rearrange making x the subject

y =
`(x+2)/(x+1)` ( multiply both sides by (x + 1)

y(x + 1) = x + 2 ← distribute left side

yx + y = x + 2 ( subtract x from both sides )

yx - x + y = 2 ( subtract y from both sides )

yx - x = 2 - y ← factor out x on the left side

x(y - 1) = 2 - y ← divide both sides by (y - 1 )

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

Change y back into terms of x, thus

`f^(-1)`(x) =
`(2-x)/(x-1)`

Answer 2

Answer:
`\bold{f^(-1)(x)=(2-x)/(x-1)\qquad \implies \qquad f^(-1)(x)=-(x-2)/(x-1)}`

Explanation:

Inverse is when you swap the x's and y's and solve for y. f(x) is y.

`y=(x+2)/(x+1)\\\\\\\\\underline{\text{Swap the x's and y's}}\\\\x=(y+2)/(y+1)\\\\\\\\\underline{\text{solve for y}}\\\\x(y+1)=y+2\\\\xy+x=y+2\qquad \qquad \text{distributed x into y+1}\\\\xy-y=2-x\qquad \qquad \text{subtracted x and y from both sides}\\\\y(x-1)=2-x\qquad \qquad \text{factored out y on the left side}\\\\\\y=(2-x)/(x-1)}\qquad \qquad \text{divided both sides by x-1}`

`\large\boxed{f^(-1)(x)=(2-x)/(x-1)}`

This is equal to:
`f^(-1)(x)=-(x-2)/(x-1)`