we are given
[tex]f(x)=\frac{1}{9x-2}[/tex]
For finding inverse function , we do following steps
step-1:
Set f(x)=y
[tex]y=\frac{1}{9x-2}[/tex]
step-2:
Switch x and y
[tex]x=\frac{1}{9y-2}[/tex]
step-3:
now, we can solve for y
[tex]x=\frac{1}{9y-2}[/tex]
Multiply both sides by 9y-2
[tex]x\times (9y-2)=(9y-2)\times \frac{1}{9y-2}[/tex]
[tex]x(9y-2)=1[/tex]
[tex]9xy-2x=1[/tex]
Add both sides by 2x
[tex]9xy-2x+2x=2x+1[/tex]
[tex]9xy=2x+1[/tex]
now, we can divide both sides by 9x
[tex]y=\frac{2x+1}{9x}[/tex]
so, we get inverse function as
[tex]f^{-1}(x)=\frac{2x+1}{9x}[/tex]................Answer
