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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−1(x)=□ \begin{array}{l} f(x)=\frac{3+4 x}{1-5 x} ? \\ f^{-1}(x)=\square \end{array}

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Q. What is the inverse of the function\newlinef(x)=3+4x1−5x?f−1(x)=□ \begin{array}{l} f(x)=\frac{3+4 x}{1-5 x} ? \\ f^{-1}(x)=\square \end{array}
  1. Switching roles and setting up the equation: To find the inverse of the function f(x)=3+4x1−5xf(x) = \frac{3+4x}{1-5x}, 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=3+4x1−5xy = \frac{3+4x}{1-5x}. Now we replace f(x)f(x) with xx and xx with yy to find the inverse function xx00: xx11.
  2. Multiplying both sides to eliminate the fraction: Next, we solve for yy in terms of xx. To do this, we multiply both sides of the equation by (1−5y)(1-5y) to get rid of the fraction:\newlinex(1−5y)=3+4y.x(1-5y) = 3+4y.
  3. Distributing and rearranging the equation: Now we distribute xx on the left side of the equation: x−5xy=3+4yx - 5xy = 3 + 4y.
  4. Isolating y and factoring out: We want to isolate y, so we'll move all terms involving y to one side and the constant to the other side: 5xy−4y=x−35xy - 4y = x - 3.
  5. Dividing both sides to solve for yy: Factor out yy from the left side of the equation:\newliney(5x−4)=x−3y(5x - 4) = x - 3.
  6. Finding the inverse function: Now, divide both sides by (5x−4)(5x - 4) to solve for yy:y=x−35x−4.y = \frac{x - 3}{5x - 4}.
  7. Finding the inverse function: Now, divide both sides by (5x−4)(5x - 4) to solve for yy:y=x−35x−4y = \frac{x - 3}{5x - 4}.We have found the inverse function. The inverse of f(x)=3+4x1−5xf(x) = \frac{3+4x}{1-5x} is f−1(x)=x−35x−4f^{-1}(x) = \frac{x - 3}{5x - 4}.

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