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$What is the inverse of the function {:[f(x)=(-2x+2)/(x+7)?],[f^(-1)(x)=◻]:}$

What is the inverse of the function $f(x)=\frac{-2x+2}{x+7}$ ? $f^{-1}(x)=\square$

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Find the vertex of the transformed function

Full solution

Q. What is the inverse of the function

f(x)=\frac{-2x+2}{x+7}

?

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

Write Function Replacement: Write down the original function and replace $f(x)$ with $y$ for convenience. $\newline$ $y = \frac{-2x + 2}{x + 7}$
Swap $x$ and $y$ : To find the inverse function, swap $x$ and $y$ in the equation. $x = \frac{-2y + 2}{y + 7}$
Solve for y: Solve for y in terms of x. Start by multiplying both sides by $(y + 7)$ to eliminate the fraction. $\newline$ $x(y + 7) = -2y + 2$
Distribute $x$ : Distribute $x$ on the left side of the equation. $xy + 7x = -2y + 2$
Move Terms: Move all terms involving $y$ to one side of the equation and the constant terms to the other side. $xy + 2y = 2 - 7x$
Factor Out $y$ : Factor out $y$ on the left side of the equation. $y(x + 2) = 2 - 7x$
Divide by $(x + 2)$ : Divide both sides by $(x + 2)$ to solve for $y$ . $y = \frac{2 - 7x}{x + 2}$
Replace with Inverse Function: Replace $y$ with $f^{-1}(x)$ to denote the inverse function. $f^{-1}(x) = \frac{2 - 7x}{x + 2}$

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