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What is the inverse of the function {:[f(x)=(5x+2)/(x-3)?],[f^(-1)(x)=]:}

What is the inverse of the function\newlinef(x)=5x+2x−3?f−1(x)=□ \begin{array}{l} f(x)=\frac{5 x+2}{x-3} ? \\ f^{-1}(x)=\square \end{array}

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Full solution

Q. What is the inverse of the function\newlinef(x)=5x+2x−3?f−1(x)=□ \begin{array}{l} f(x)=\frac{5 x+2}{x-3} ? \\ f^{-1}(x)=\square \end{array}
  1. Switching Roles: To find the inverse of the function f(x)=5x+2x−3f(x) = \frac{5x+2}{x-3}, 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=5x+2x−3y = \frac{5x+2}{x-3}. Now we switch xx and yy to find the inverse: x=5y+2y−3x = \frac{5y+2}{y-3}.
  2. Solving for y: Next, we solve for y. To do this, we'll multiply both sides of the equation by (y−3)(y-3) to get rid of the fraction:\newlinex(y−3)=5y+2x(y-3) = 5y + 2.
  3. Distributing xx: Distribute xx on the left side of the equation:\newlinexy−3x=5y+2xy - 3x = 5y + 2.
  4. Moving Terms: Now, we want to get all the terms with yy on one side and the constants on the other side. So, we'll move the 5y5y term to the left side by subtracting 5y5y from both sides: xy−5y=3x+2xy - 5y = 3x + 2.
  5. Factoring out yy: Factor out yy from the left side of the equation:\newliney(x−5)=3x+2y(x - 5) = 3x + 2.
  6. Dividing Both Sides: Finally, divide both sides by (x−5)(x - 5) to solve for yy:y=3x+2x−5.y = \frac{3x + 2}{x - 5}.This is the inverse function of f(x)f(x).

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