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For the function f(x)=(10+3x)/(1-4x), find f^(-1)(x). Answer: f^(-1)(x)=

For the function f(x)=10+3x1−4x f(x)=\frac{10+3 x}{1-4 x} , find f−1(x) f^{-1}(x) .\newlineAnswer: f−1(x)= f^{-1}(x)=

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Q. For the function f(x)=10+3x1−4x f(x)=\frac{10+3 x}{1-4 x} , find f−1(x) f^{-1}(x) .\newlineAnswer: f−1(x)= f^{-1}(x)=
  1. Replace with y: To find the inverse function, f−1(x)f^{-1}(x), we need to switch the roles of xx and yy in the original function and then solve for yy. Let's start by replacing f(x)f(x) with yy:y=10+3x1−4xy = \frac{10 + 3x}{1 - 4x}
  2. Interchange x and y: Now, interchange x and y to find the inverse: x=10+3y1−4yx = \frac{10 + 3y}{1 - 4y}
  3. Eliminate the fraction: Next, we need to solve for yy. Start by multiplying both sides of the equation by (1−4y)(1 - 4y) to eliminate the fraction:\newlinex⋅(1−4y)=10+3yx \cdot (1 - 4y) = 10 + 3y
  4. Move terms around: Distribute xx on the left side of the equation:\newlinex−4xy=10+3yx - 4xy = 10 + 3y
  5. Factor out yy: Now, we want to get all the terms with yy on one side and the constant terms on the other side. Let's move the 3y3y term to the left side and the xx term to the right side:\newline4xy+3y=10−x4xy + 3y = 10 - x
  6. Divide to solve for y: Factor out y from the left side of the equation:\newliney(4x+3)=10−xy(4x + 3) = 10 - x
  7. Final Inverse Function: Now, divide both sides by (4x+3)(4x + 3) to solve for yy:y=10−x4x+3y = \frac{10 - x}{4x + 3}
  8. Final Inverse Function: Now, divide both sides by (4x+3)(4x + 3) to solve for yy:y=10−x4x+3y = \frac{10 - x}{4x + 3}We have found the inverse function. Therefore, the inverse function f−1(x)f^{-1}(x) is:f−1(x)=10−x4x+3f^{-1}(x) = \frac{10 - x}{4x + 3}

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