Questions: Find the inverse of the following function. f(x) = (-3x + 5) / (7x + 4)

Transcript text: Question Find the inverse of the following function. \[ f(x)=\frac{-3 x+5}{7 x+4} \] Provide your answer below: \[ f^{-1}(x)= \]

Solution

Solution Steps

To find the inverse of a function, we need to swap the roles of \(x\) and \(y\) in the equation and then solve for \(y\). For the given function \(f(x) = \frac{-3x + 5}{7x + 4}\), we will set \(y = \frac{-3x + 5}{7x + 4}\), swap \(x\) and \(y\), and solve for \(y\) to find the inverse function.

Step 1: Define the Function

We start with the function given by \[ f(x) = \frac{-3x + 5}{7x + 4}. \]

Step 2: Set Up the Inverse

To find the inverse, we set \(y = f(x)\): \[ y = \frac{-3x + 5}{7x + 4}. \] Next, we swap \(x\) and \(y\): \[ x = \frac{-3y + 5}{7y + 4}. \]

Step 3: Solve for \(y\)

We rearrange the equation to solve for \(y\): \[ x(7y + 4) = -3y + 5. \] Expanding and rearranging gives: \[ 7xy + 4x = -3y + 5. \] Bringing all terms involving \(y\) to one side results in: \[ 7xy + 3y = 5 - 4x. \] Factoring out \(y\) yields: \[ y(7x + 3) = 5 - 4x. \] Finally, we solve for \(y\): \[ y = \frac{5 - 4x}{7x + 3}. \]