Finding the inverse function for the following function

Question

Find −1() for each function.

My answer:

or

The book answer:

I don't understand how the book concluded with this answer?

Answer 1

I'm not clear on how you got the first line in your answer,

$$x=y \dfrac{x}{x+2}$$

but it shows some flaw in your understanding of inverses.

In an inverse function, the dependent and independent variables (i.e. $x$ and $y$) switch roles. This means that to find an inverse algebraically, the first step is to replace each $y$ with an $x$ and each $x$ with a $y$. Thus your first line should be

$$x=\dfrac{y}{y+2}$$

Then, assuming the equation is not too complex, one solves for y. You should try this for yourself first, but I've included the solution below in case you run into trouble.

$$x=\dfrac{y}{y+2}$$ $$x(y+2)=y$$ $$xy+2x=y$$ $$xy-y=-2x$$ $$y(x-1)=-2x$$ $$y=\dfrac{-2x}{x-1}$$