Question

Find the inverse of the function f(x)=5x-6/x+2

Answer 1

Answer: The required inverse function is

`g(x)=(2x+6)/(5-x).`

Step-by-step explanation: We are given to find the inverse of the following function :

`f(x)=(5x-6)/(x+2)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(i)`

Let us consider that

`y=f(x)~~~~\Rightarrow f^(-1)(y)=x.`

From equation (i), we get

`f(x)=(5x-6)/(x+2)\\\\\\\Rightarrow y=(5f^(-1)(y)-6)/(f^(-1)(y)+2)\\\\\\\Rightarrow yf^(-1)(y)+2y=5f^(-1)(y)-6\\\\\\\Rightarrow (y-5)f^(-1)(y)=-2y-6\\\\\\\Rightarrow f^(-1)(y)=(2y+6)/(5-y).`

Thus, the required inverse function is

`g(x)=(2x+6)/(5-x).`

Answer 2

`f^(-1)(x)=(2x+6)/(5-x)`

Explanation:

Swap x and y, then solve for y.

`x=(5y-6)/(y+2)\\\\x(y+2)=5y-6\\\\xy+2x=5y-6\\\\2x+6=5y-xy\\\\y=(2x+6)/(5-x)\\\\f^(-1)(x)=(2x+6)/(5-x)`