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What is the inverse of the function {:[f(x)=(6x-5)/(x+9)?],[f^(-1)(x)=]:}

What is the inverse of the function\newlinef(x)=6x−5x+9?f−1(x)=□ \begin{array}{l} f(x)=\frac{6 x-5}{x+9} ? \\ f^{-1}(x)=\square \end{array}

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Full solution

Q. What is the inverse of the function\newlinef(x)=6x−5x+9?f−1(x)=□ \begin{array}{l} f(x)=\frac{6 x-5}{x+9} ? \\ f^{-1}(x)=\square \end{array}
  1. Switching roles and solving for xx: To find the inverse of the function f(x)=6x−5x+9f(x) = \frac{6x - 5}{x + 9}, we need to switch the roles of xx and f(x)f(x) and then solve for the new xx. Let y=f(x)y = f(x), so we have y=6x−5x+9y = \frac{6x - 5}{x + 9}. Now, we replace yy with xx to get x=6y−5y+9x = \frac{6y - 5}{y + 9}.
  2. Eliminating the denominator: Next, we solve for yy by multiplying both sides of the equation by (y+9)(y + 9) to eliminate the denominator.\newlinex(y+9)=6y−5x(y + 9) = 6y - 5\newlinexy+9x=6y−5xy + 9x = 6y - 5
  3. Rearranging the equation: Now, we need to get all the terms with yy on one side and the constants on the other side.xy−6y=−5−9xxy - 6y = -5 - 9xy(x−6)=−5−9xy(x - 6) = -5 - 9x
  4. Isolating yy: To isolate yy, we divide both sides of the equation by (x−6)(x - 6).\newliney=−5−9xx−6y = \frac{-5 - 9x}{x - 6}
  5. Expressing the inverse function: Finally, we express the inverse function as f−1(x)f^{-1}(x). f−1(x)=−5−9xx−6f^{-1}(x) = \frac{-5 - 9x}{x - 6}

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