Question

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

Answer 1

Answer:

  $f^{-1}(x)=\dfrac{2x+6}{5-x}$

Step-by-step explanation:

Swap x and y, then solve for y.

$x=\dfrac{5y-6}{y+2}\\\\x(y+2)=5y-6\\\\xy+2x=5y-6\\\\2x+6=5y-xy\\\\y=\dfrac{2x+6}{5-x}\\\\f^{-1}(x)=\dfrac{2x+6}{5-x}$

Answer 2

Answer:  The required inverse function is

$g(x)=\dfrac{2x+6}{5-x}.$

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

$f(x)=\dfrac{5x-6}{x+2}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(i)$

Let us consider that

$y=f(x)~~~~\Rightarrow f^{-1}(y)=x.$

From equation (i), we get

$f(x)=\dfrac{5x-6}{x+2}\\\\\\\Rightarrow y=\dfrac{5f^{-1}(y)-6}{f^{-1}(y)+2}\\\\\\\Rightarrow yf^{-1}(y)+2y=5f^{-1}(y)-6\\\\\\\Rightarrow (y-5)f^{-1}(y)=-2y-6\\\\\\\Rightarrow f^{-1}(y)=\dfrac{2y+6}{5-y}.$

Thus, the required inverse function is

$g(x)=\dfrac{2x+6}{5-x}.$