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

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

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Identify inverse functions

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Q. What is the inverse of the function

\newline

\begin{array}{l} h(x)=\frac{5 x-3}{x-1} ? \\ h^{-1}(x)=\square \end{array}

Replace with y: To find the inverse of the function $h(x)$ , we need to switch the roles of $x$ and $y$ in the equation and then solve for $y$ . Let's start by replacing $h(x)$ with $y$ : $y = \frac{5x - 3}{x - 1}$
Switch x and y: Now we switch x and y to find the inverse: $x = \frac{5y - 3}{y - 1}$
Multiply and solve: Next, we solve for $y$ . Multiply both sides by $(y - 1)$ to get rid of the fraction: $\newline$ $x(y - 1) = 5y - 3$
Distribute $x$ : Distribute $x$ on the left side of the equation: $\newline$ $xy - x = 5y - 3$
Move $5y$ term: Now, we want to get all the terms with $y$ on one side and the constants on the other. Let's move the $5y$ term to the left side by subtracting $5y$ from both sides: $xy - 5y = x + 3$
Factor out $y$ : Factor out $y$ on the left side of the equation: $\newline$ $y(x - 5) = x + 3$
Divide and solve: Finally, divide both sides by $(x - 5)$ to solve for $y$ : $y = \frac{x + 3}{x - 5}$
Final Inverse Function: We have found the inverse function of $h(x)$ . The inverse function is: $\newline$ $h^{-1}(x) = \frac{x + 3}{x - 5}$

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