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

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Q. What is the inverse of the function\newlineg(x)=9x+4x−7?g−1(x)=□ \begin{array}{l} g(x)=\frac{9 x+4}{x-7} ? \\ 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=9x+4x−7y = \frac{9x + 4}{x - 7}
  2. Switch x and y: Now, we switch x and y to find the inverse function:\newlinex=9y+4y−7x = \frac{9y + 4}{y - 7}
  3. Eliminate the denominator: Next, we solve for yy. Multiply both sides by (y−7)(y - 7) to eliminate the denominator: x(y−7)=9y+4x(y - 7) = 9y + 4
  4. Distribute xx: Distribute xx on the left side of the equation:\newlinexy−7x=9y+4xy - 7x = 9y + 4
  5. 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 9y9y term to the left side by subtracting 9y9y from both sides:\newlinexy−9y−7x=4xy - 9y - 7x = 4
  6. Factor out y: Factor out y from the left side:\newliney(x−9)−7x=4y(x - 9) - 7x = 4
  7. Isolate y: Now, isolate y by adding 7x7x to both sides and then dividing by (x−9)(x - 9):\newliney=4+7xx−9y = \frac{4 + 7x}{x - 9}
  8. Inverse function found: We have found the inverse function, which we can denote as g−1(x)g^{-1}(x):\newlineg−1(x)=4+7xx−9g^{-1}(x) = \frac{4 + 7x}{x - 9}

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