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$What is the inverse of the function {:[g(x)=(9x+4)/(x-7)?],[g^(-1)(x)=◻]:}$

What is the inverse of the function $g(x) = \frac{9x + 4}{x - 7}$ ? $g^{-1}(x) = \square$

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Find the vertex of the transformed function

Full solution

Q. What is the inverse of the function

g(x) = \frac{9x + 4}{x - 7}

?

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

Write Function as y: Write down the function $g(x)$ and replace $g(x)$ with $y$ for convenience. $\newline$ $y = \frac{9x + 4}{x - 7}$
Switch Roles of x and y: To find the inverse function, we need to switch the roles of $x$ and $y$ . This means we will replace $y$ with $x$ and $x$ with $y$ in the equation. $\newline$ $x = \frac{9y + 4}{y - 7}$
Solve for y: Solve the equation from Step $2$ for y. Start by multiplying both sides by $(y - 7)$ to eliminate the fraction. $\newline$ $x(y - 7) = 9y + 4$
Distribute $x$ : Distribute $x$ on the left side of the equation. $xy - 7x = 9y + 4$
Combine Like Terms: Get all terms containing $y$ on one side and the constant terms on the other side. $xy - 9y = 7x + 4$
Factor Out $y$ : Factor out $y$ from the left side of the equation. $y(x - 9) = 7x + 4$
Solve for y: Divide both sides by $(x - 9)$ to solve for y. $\newline$ $y = \frac{7x + 4}{x - 9}$

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