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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 f(x)=−2x+2x+7 f(x)=\frac{-2x+2}{x+7} ? f−1(x)=□ f^{-1}(x)=\square

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Full solution

Q. What is the inverse of the function f(x)=−2x+2x+7 f(x)=\frac{-2x+2}{x+7} ? f−1(x)=□ f^{-1}(x)=\square
  1. Write Function Replacement: Write down the original function and replace f(x)f(x) with yy for convenience.\newliney=−2x+2x+7y = \frac{-2x + 2}{x + 7}
  2. Swap xx and yy: To find the inverse function, swap xx and yy in the equation.x=−2y+2y+7x = \frac{-2y + 2}{y + 7}
  3. Solve for y: Solve for y in terms of x. Start by multiplying both sides by (y+7)(y + 7) to eliminate the fraction.\newlinex(y+7)=−2y+2x(y + 7) = -2y + 2
  4. Distribute xx: Distribute xx on the left side of the equation.xy+7x=−2y+2xy + 7x = -2y + 2
  5. Move Terms: Move all terms involving yy to one side of the equation and the constant terms to the other side.xy+2y=2−7xxy + 2y = 2 - 7x
  6. Factor Out yy: Factor out yy on the left side of the equation.y(x+2)=2−7xy(x + 2) = 2 - 7x
  7. Divide by (x+2)(x + 2): Divide both sides by (x+2)(x + 2) to solve for yy.y=2−7xx+2y = \frac{2 - 7x}{x + 2}
  8. Replace with Inverse Function: Replace yy with f−1(x)f^{-1}(x) to denote the inverse function.f−1(x)=2−7xx+2f^{-1}(x) = \frac{2 - 7x}{x + 2}

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