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Understanding one-to-one functions is crucial in finding inverses. A function is one-to-one if every output value is paired with a unique input value. This means that if \(f(a) = f(b)\), then \(a = b\). To check if a function is one-to-one, you can use the horizontal line test: if any horizontal line intersects the graphed function at most once, the function is one-to-one.

To find the inverse of a function, you need to solve for the variable. Starting from the given function \(f(x) = \frac{-3x + 2}{3x - 4}\), substitute \(f(x)\) with \y \, resulting in \(y = \frac{-3x + 2}{3x - 4}\). Interchanging \(x \) and \(y \), gives \(x = \frac{-3y + 2}{3y - 4}\). From here, you need to isolate \ y\. Multiply both sides by \(3y - 4\) to clear the fraction: \[ x(3y - 4) = -3y + 2 \] Distribute \(x \) and rearrange the terms to: \[3xy - 4x = -3y + 2\] Move the \( y\) terms to one side: \[3xy + 3y = 2 + 4x\] Factor out \(y \) to get: \[y(3x + 3) = 2 + 4x\] Finally, solve for \ y\: \[y = \frac{2 + 4x}{3x + 3}\] Thus, the inverse function is \(f^{-1}(x) = \frac{2 + 4x}{3x + 3}\).

Finding the inverse of a function involves reversing the roles of inputs and outputs. For the function \(f(x) = \frac{-3x + 2}{3x - 4}\), to find its inverse follow these steps:

• Step 1: Write \(f(x)\) as \(y = \frac{-3x + 2}{3x - 4}\).
• Step 2: Swap \(x \) and \(y \) to get \ x = \frac{-3y + 2}{3y - 4} \.
• Step 3: Solve for \( y\).

By solving for \ y\, we find the inverse function to be \( f^{-1}(x) = \frac{2 + 4x}{3x + 3} \). An important property of inverse functions is that if you have \(f^{-1}(f(x)) = x \) and \(f(f^{-1}(x)) = x \), then the functions are indeed inverses of each other.