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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 $\newline$ $\begin{array}{l} g(x)=\frac{9 x+4}{x-7} ? \\ g^{-1}(x)=\square \end{array}$

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Identify inverse functions

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Q. What is the inverse of the function

\newline

\begin{array}{l} g(x)=\frac{9 x+4}{x-7} ? \\ g^{-1}(x)=\square \end{array}

Replace $g(x)$ with $y$ : To find the inverse of the function $g(x)$ , we need to switch the roles of $x$ and $y$ in the equation and then solve for $y$ . Let's start by replacing $g(x)$ with $y$ : $y = \frac{9x + 4}{x - 7}$
Switch x and y: Now, we switch x and y to find the inverse function: $\newline$ $x = \frac{9y + 4}{y - 7}$
Eliminate the denominator: Next, we solve for $y$ . Multiply both sides by $(y - 7)$ to eliminate the denominator: $x(y - 7) = 9y + 4$
Distribute $x$ : Distribute $x$ on the left side of the equation: $\newline$ $xy - 7x = 9y + 4$
Move terms with y to one side: To isolate y, we need to get all the terms with y on one side and the constants on the other. Let's move the $9y$ term to the left side by subtracting $9y$ from both sides: $\newline$ $xy - 9y - 7x = 4$
Factor out y: Factor out y from the left side: $\newline$ $y(x - 9) - 7x = 4$
Isolate y: Now, isolate y by adding $7x$ to both sides and then dividing by $(x - 9)$ : $\newline$ $y = \frac{4 + 7x}{x - 9}$
Inverse function found: We have found the inverse function, which we can denote as $g^{-1}(x)$ : $\newline$ $g^{-1}(x) = \frac{4 + 7x}{x - 9}$

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