Need help with the following: if `f(x) = 2x-3;` `g(x) = 1/(x+1),` find `(f@g)^(-1)(x).`

Expert Answers

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First found the composition:


This function is defined everywhere except `x=-1.`


To find the inverse function, we have to solve for `x` the equation `(f@g)(x)=y:`

`-(3x+1)/(x+1)=y,`  or `(3x+1)/(x+1)=-y.`


Multiply both sides by `(x+1)` and obtain



Move the terms with `x` to the left and without `x` to the right:


so  `x=-(y+1)/(y+3)`  (of course `y!=-3` ).


Thus the answer is: the function `(f@g)^(-1)(y)` exists for all `y!=-3` and is equal to `-(y+1)/(y+3).`

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