QuestionFind the inverse of the function $f(x)=\frac{4x-1}{6x-9}$ .

Studdy Solution

STEP 1

Assumptions1. The function is $f(x)=\frac{4 x-1}{6 x-9}$ . The function $f(x)$ is invertible, meaning there exists a function $f^{-1}(x)$ such that $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$

STEP 2

To find the inverse of a function, we first replace $f(x)$ with $y$ . This gives us $y=\frac{4 x-1}{6 x-9}$

STEP 3

The next step in finding the inverse of a function is to switch $x$ and $y$ . This gives us $x=\frac{ y-1}{6 y-9}$

STEP 4

Now, we need to solve this equation for $y$ . First, we cross multiply to get rid of the fraction $x(6y -9) =4y -1$

STEP 5

istribute the $x$ on the left side of the equation $xy -9x =4y -1$

STEP 6

Rearrange the equation to group terms with $y$ on one side and constants on the other $6xy -4y =1 +9x$

STEP 7

Factor out $y$ from the left side of the equation $y(6x -4) =1 +9x$

STEP 8

Finally, divide both sides of the equation by $(6x -4)$ to solve for $y$ :
$y = \frac{1 +x}{6x -4}$ This is the inverse of the original function, $f(x)$ . So, $f^{-1}(x) = \frac{1 +x}{6x -4}$ .

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