Answer:
Inverse function is: [tex]g^{-1}x=\frac{-x\pm\sqrt{x(x+3)}}{x}[/tex]
Step-by-step explanation:
Given Given the function [tex]g(x)=\frac{3}{x^2+2x}[/tex]
we have to find the inverse function of g
To find inverse following steps are:
Step 1: Replace g(x) with y
Step 2: Replace every x with a y and every y with x
Step 3: Solve the equation obtained from Step 2 for y.
Step 4: Replace y with g−1(x).
[tex]g(x)=\frac{3}{x^2+2x}[/tex]
To find the inverse, let [tex]y=\frac{3}{x^2+2x}[/tex]
Now, solve for x
[tex]y(x^2+2x)=3[/tex]
[tex]yx^2+2yx-3=0[/tex]
[tex]x=\frac{-2y\pm\sqrt{4y^2-4y(-3)}}{2y}[/tex]
[tex]x=\frac{-2y\pm\sqrt{4y^2+12y}}{2y}[/tex]
[tex]x=\frac{-2y\pm2\sqrt{y^2+3y}}{2y}[/tex]
[tex]x=\frac{-y\pm\sqrt{y(y+3)}}{y}[/tex]
Replace every x with a y and every y with x , we get
[tex]y=\frac{-x\pm\sqrt{x(x+3)}}{x}[/tex]
Replace y with g−1(x).
[tex]g^{-1}x=\frac{-x\pm\sqrt{x(x+3)}}{x}[/tex]
So inverse function is: [tex]g^{-1}x=\frac{-x\pm\sqrt{x(x+3)}}{x}[/tex]
