Question

Find g^-1(x)

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B
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D

Answer 1

`a. \ g^(-1)=-1 \pm\sqrt{(3)/(x)+1}`

Explanation:

A function has an inverse function if and only if passes the Horizontal Line Test for Inverse Functions. This test tells us that a function
`f` has an inverse function if and only if there is no any horizontal line that intersects the graph of
`f` at more than one point. So the function is called one-to-one. The graph of
`g` is shown below. As you can see, this function does not pass the Horizontal Line Test, therefore the inverse is not a function. However, let's find
`g^-{1}(x)`:

`g(x)=(3)/(x^2+2x) \\ \\ Substitute \ g(x) \ by \ y \\ \\y=(3)/(x^2+2x) \\ \\ Interchange \ x \ and \ y: \\ \\ x=(3)/(y^2+2y) \\ \\ Solve \ for \ y: \\ \\y^2+2y=(3)/(x) \\ \\ Completing \ square \\ \\y^2+2y\mathbf{+ 1}\mathbf{-1}=(3)/(x) \\ \\(y+1)^2=(3)/(x)+1 \\ \\y+1=\pm\sqrt{(3)/(x)+1} \\ \\ y=-1 \pm\sqrt{(3)/(x)+1}`

Finally, substitute
`y \ by \ g^(-1)`:

`boxed{g^(-1)=-1 \pm\sqrt{(3)/(x)+1}}`