QuestionFind the inverse function $f^{-1}(x)$ for $f(x)=\frac{2x+3}{x-5}$ .

Studdy Solution

STEP 1

Assumptions1. The function $f(x)$ is given as $f(x)=\frac{ x+3}{x-5}$ . We are tasked to find the inverse of the function, denoted as $f^{-1}(x)$

STEP 2

The first step to find the inverse of a function is to replace $f(x)$ with $y$ . This gives us $y = \frac{2 x+}{x-5}$

STEP 3

Next, we swap $x$ and $y$ . This means we replace every $x$ in our equation with $y$ and vice versa. This gives us $x = \frac{2 y+3}{y-5}$

STEP 4

Now, we solve this equation for $y$ to find $f^{-1}(x)$ . First, we multiply both sides by $y-$ to get rid of the denominator on the right side $(y-)x =2y +3$

STEP 5

Next, we distribute $x$ on the left side $yx -5x =2y +3$

STEP 6

We then move the terms involving $y$ to one side and the rest to the other side $yx -2y =5x +3$

STEP 7

We factor out $y$ on the left side $y(x -2) =5x +3$

STEP 8

Finally, we divide both sides by $x -2$ to solve for $y$ $y = \frac{5x +3}{x -2}$

STEP 9

We replace $y$ with $f^{-}(x)$ to get the inverse function $f^{-}(x) = \frac{5x +3}{x -2}$ So, the inverse of the function $f(x)=\frac{2 x+3}{x-5}$ is $f^{-}(x) = \frac{5x +3}{x -2}$ .

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