Answer:
[tex]a. \ g^{-1}=-1 \pm\sqrt{\frac{3}{x}+1}[/tex]
Step-by-step 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 [tex]f[/tex] has an inverse function if and only if there is no any horizontal line that intersects the graph of [tex]f[/tex] at more than one point. So the function is called one-to-one. The graph of [tex]g[/tex] 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 [tex]g^-{1}(x)[/tex]:
[tex]g(x)=\frac{3}{x^2+2x} \\ \\ Substitute \ g(x) \ by \ y \\ \\y=\frac{3}{x^2+2x} \\ \\ Interchange \ x \ and \ y: \\ \\ x=\frac{3}{y^2+2y} \\ \\ Solve \ for \ y: \\ \\y^2+2y=\frac{3}{x} \\ \\ Completing \ square \\ \\y^2+2y\mathbf{+ 1}\mathbf{-1}=\frac{3}{x} \\ \\(y+1)^2=\frac{3}{x}+1 \\ \\y+1=\pm\sqrt{\frac{3}{x}+1} \\ \\ y=-1 \pm\sqrt{\frac{3}{x}+1}[/tex]
Finally, substitute [tex]y \ by \ g^{-1}[/tex]:
[tex]boxed{g^{-1}=-1 \pm\sqrt{\frac{3}{x}+1}}[/tex]
