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$What is the inverse of the function {:[g(x)=(2x-1)/(x+3)?],[g^(-1)(x)=]:}$

What is the inverse of the function $\newline$ $\begin{array}{l} g(x)=\frac{2 x-1}{x+3} ? \\ g^{-1}(x)= \end{array}$

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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{2 x-1}{x+3} ? \\ g^{-1}(x)= \end{array}

Replace $g(x)$ with $y$ : To find the inverse of the function $g(x) = \frac{2x - 1}{x + 3}$ , we need to switch the roles of $x$ and $y$ and then solve for $y$ . Let's start by replacing $g(x)$ with $y$ : $y = \frac{2x - 1}{x + 3}$
Switch x and y: Now we switch x and y to find the inverse: $\newline$ $x = \frac{2y - 1}{y + 3}$
Multiply both sides by $(y + 3)$ : Next, we solve for $y$ . To do this, we'll multiply both sides of the equation by $(y + 3)$ to eliminate the denominator: $\newline$ $x(y + 3) = 2y - 1$
Distribute $x$ on the left side: Distribute $x$ on the left side of the equation: $xy + 3x = 2y - 1$
Move terms involving $y$ to the left side: To isolate $y$ , we need to get all the terms with $y$ on one side and the constant terms on the other side. Let's move the terms involving $y$ to the left side and the constant terms to the right side: $\newline$ $xy - 2y = -1 - 3x$
Factor out $y$ : Factor out $y$ from the left side of the equation: $\newline$ $y(x - 2) = -1 - 3x$
Divide both sides by $(x - 2)$ : Now, divide both sides by $(x - 2)$ to solve for $y$ : $y = \frac{-1 - 3x}{x - 2}$
Simplify the right side: We can simplify the right side of the equation by distributing the negative sign: $\newline$ $y = \frac{-1 - 3x}{x - 2} = \frac{-1}{x - 2} - \frac{3x}{x - 2}$
Inverse function found: The expression is already simplified, so we have found the inverse function: $g^{-1}(x) = -\frac{1}{x - 2} - \frac{3x}{x - 2}$

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