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What is the inverse of the function {:[f(x)=(-2x+2)/(x+7)?],[f^(-1)(x)=]:}

What is the inverse of the function\newlinef(x)=−2x+2x+7?f−1(x)=□ \begin{array}{l} f(x)=\frac{-2 x+2}{x+7} ? \\ f^{-1}(x)=\square \end{array}

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Q. What is the inverse of the function\newlinef(x)=−2x+2x+7?f−1(x)=□ \begin{array}{l} f(x)=\frac{-2 x+2}{x+7} ? \\ f^{-1}(x)=\square \end{array}
  1. Switching Roles: To find the inverse of the function f(x)f(x), 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=−2x+2x+7y = \frac{-2x + 2}{x + 7} Now we switch xx and yy: x=−2y+2y+7x = \frac{-2y + 2}{y + 7}
  2. Multiplying to Eliminate Denominator: Next, we need to solve for yy. To do this, we'll multiply both sides of the equation by (y+7)(y + 7) to eliminate the denominator:\newlinex(y+7)=−2y+2x(y + 7) = -2y + 2\newlinexy+7x=−2y+2xy + 7x = -2y + 2
  3. Moving Terms to Isolate y: Now, we'll move all terms involving y to one side and the constant terms to the other side:\newlinexy+2y=2−7xxy + 2y = 2 - 7x\newliney(x+2)=2−7xy(x + 2) = 2 - 7x
  4. Dividing to Isolate y: To isolate y, we divide both sides by (x+2)(x + 2):\newliney=2−7xx+2y = \frac{2 - 7x}{x + 2}
  5. Inverse Function: This gives us the inverse function of f(x)f(x):f−1(x)=2−7xx+2f^{-1}(x) = \frac{2 - 7x}{x + 2}

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