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$What is the inverse of the function {:[f(x)=(6x-5)/(x+9)" ? "],[f^(-1)(x)=◻]:}$

What is the inverse of the function $f(x) = \frac{6x - 5}{x + 9}$ ? $f^{-1}(x) = \square$

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Find the vertex of the transformed function

Full solution

Q. What is the inverse of the function

f(x) = \frac{6x - 5}{x + 9}

?

f^{-1}(x) = \square

Replace with $y$ : To find the inverse of the function $f(x) = \frac{6x - 5}{x + 9}$ , we first replace $f(x)$ with $y$ for simplicity. $\newline$ $y = \frac{6x - 5}{x + 9}$
Interchange $x$ and $y$ : Next, we interchange the roles of $x$ and $y$ to find the inverse. This means we replace $y$ with $x$ and $x$ with $y$ in the equation. $x = \frac{6y - 5}{y + 9}$
Solve for y: Now, we solve for y. To do this, we first multiply both sides of the equation by $(y + 9)$ to eliminate the denominator. $x(y + 9) = 6y - 5$
Multiply by $(y + 9)$ : Distribute $x$ on the left side of the equation. $xy + 9x = 6y - 5$
Distribute $x$ : We want to isolate $y$ , so we move all terms involving $y$ to one side and the constant terms to the other side. $xy - 6y = -5 - 9x$
Isolate y: Factor out y from the left side of the equation. $\newline$ $y(x - 6) = -5 - 9x$
Factor out $y$ : Divide both sides by $(x - 6)$ to solve for $y$ . $\newline$ $y = \frac{-5 - 9x}{x - 6}$
Divide by $(x - 6)$ : This is the inverse function, so we replace $y$ with $f^{-1}(x)$ . $\newline$ $f^{-1}(x) = \frac{-5 - 9x}{x - 6}$

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