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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\newlineg(x)=7x+3x−5?g−1(x)=□ \begin{array}{l} g(x)=\frac{7 x+3}{x-5} ? \\ g^{-1}(x)=\square \end{array}

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Full solution

Q. What is the inverse of the function\newlineg(x)=7x+3x−5?g−1(x)=□ \begin{array}{l} g(x)=\frac{7 x+3}{x-5} ? \\ g^{-1}(x)=\square \end{array}
  1. Replace g(x)g(x) with yy: To find the inverse of the function g(x)g(x), we need to switch the roles of xx and yy in the equation and then solve for yy. Let's start by replacing g(x)g(x) with yy:y=7x+3x−5y = \frac{7x + 3}{x - 5}
  2. Switch x and y: Now we switch x and y to find the inverse: x=7y+3y−5x = \frac{7y + 3}{y - 5}
  3. Eliminate the fraction: Next, we solve for yy. Multiply both sides by (y−5)(y - 5) to eliminate the fraction: x(y−5)=7y+3x(y - 5) = 7y + 3
  4. Move terms to one side: Distribute xx on the left side of the equation: xy−5x=7y+3xy - 5x = 7y + 3
  5. Factor out yy: To solve for yy, we need to get all the yy terms on one side and the constants on the other. Let's move the 7y7y term to the left side by subtracting 7y7y from both sides:\newlinexy−7y−5x=3xy - 7y - 5x = 3
  6. Isolate yy: Factor out yy from the left side of the equation:\newliney(x−7)−5x=3y(x - 7) - 5x = 3
  7. Solve for yy: Now, isolate yy by adding 5x5x to both sides:\newliney(x−7)=5x+3y(x - 7) = 5x + 3
  8. Solve for y: Now, isolate y by adding 5x5x to both sides:\newliney(x−7)=5x+3y(x - 7) = 5x + 3 Finally, divide both sides by (x−7)(x - 7) to solve for y:\newliney=5x+3x−7y = \frac{5x + 3}{x - 7}

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