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What is the inverse of the function {:[f(x)=(3+4x)/(1-5x)?],[f^(-1)(x)=◻]:}

What is the inverse of the function\newlinef(x)=3+4x1−5x f(x) = \frac{3 + 4x}{1 - 5x} ?\newlinef−1(x)=□ f^{-1}(x) = \square

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Full solution

Q. What is the inverse of the function\newlinef(x)=3+4x1−5x f(x) = \frac{3 + 4x}{1 - 5x} ?\newlinef−1(x)=□ f^{-1}(x) = \square
  1. Understand Inverse Function Concept: Understand the concept of finding an inverse function. To find the inverse of a function, we swap the xx and yy and then solve for yy. This gives us the inverse function, denoted as f−1(x)f^{-1}(x).
  2. Write Original Function: Write the original function with yy instead of f(x)f(x).y=3+4x1−5xy = \frac{3+4x}{1-5x}
  3. Swap x and y: Swap x and y to begin finding the inverse.\newliney=3+4x1−5xy = \frac{3+4x}{1-5x}
  4. Solve for y: Solve for y by multiplying both sides by (1−5y)(1-5y) to eliminate the fraction.\newlinex(1−5y)=3+4yx(1-5y) = 3+4y
  5. Distribute xx: Distribute xx on the left side of the equation.x−5xy=3+4yx - 5xy = 3 + 4y
  6. Combine Like Terms: Get all terms containing yy on one side and the constant terms on the other side.5xy−4y=3−x5xy - 4y = 3 - x
  7. Factor Out yy: Factor out yy from the left side of the equation.y(5x−4)=3−xy(5x - 4) = 3 - x
  8. Divide to Solve for y: Divide both sides by (5x−4)(5x - 4) to solve for yy.y=3−x5x−4y = \frac{3 - x}{5x - 4}
  9. Write Inverse Function: Write the inverse function. f−1(x)=3−x5x−4f^{-1}(x) = \frac{3 - x}{5x - 4}

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