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$What is the inverse of the function {:[g(x)=(7x+3)/(x-5)?],[g^(-1)(x)=]:}$

What is the inverse of the function $\newline$ $\begin{array}{l} g(x)=\frac{7 x+3}{x-5} ? \\ g^{-1}(x)=\square \end{array}$

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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{7 x+3}{x-5} ? \\ 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 $y$ : $y = \frac{7x + 3}{x - 5}$
Switch x and y: Now we switch x and y to find the inverse: $x = \frac{7y + 3}{y - 5}$
Eliminate the fraction: Next, we solve for $y$ . Multiply both sides by $(y - 5)$ to eliminate the fraction: $x(y - 5) = 7y + 3$
Move terms to one side: Distribute $x$ on the left side of the equation: $xy - 5x = 7y + 3$
Factor out $y$ : To solve for $y$ , we need to get all the $y$ terms on one side and the constants on the other. Let's move the $7y$ term to the left side by subtracting $7y$ from both sides: $\newline$ $xy - 7y - 5x = 3$
Isolate $y$ : Factor out $y$ from the left side of the equation: $\newline$ $y(x - 7) - 5x = 3$
Solve for $y$ : Now, isolate $y$ by adding $5x$ to both sides: $\newline$ $y(x - 7) = 5x + 3$
Solve for y: Now, isolate y by adding $5x$ to both sides: $\newline$ $y(x - 7) = 5x + 3$ Finally, divide both sides by $(x - 7)$ to solve for y: $\newline$ $y = \frac{5x + 3}{x - 7}$

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