Find the inverse \( f^{\prime} \) of the function \( f(x)=3 x+4 \) \( \begin{array}{llll}\text { A) } f(x)=4 x-3 & \text { 6) } f(x)=x-4 / 3 & \text { C) } f^{-1}(x)=\frac{x-4}{3} & \text { D) } f^{-1}(x)=\frac{x+4}{3} \\ \text { E) None of These }\end{array} \)

To find the inverse of the function \( f(x) = 3x + 4 \), we will follow these steps: 1. Start by replacing \( f(x) \) with \( y \): \[ y = 3x + 4 \] 2. Solve for \( x \) in terms of \( y \): \[ y - 4 = 3x \implies x = \frac{y - 4}{3} \] 3. Replace \( y \) with \( x \) to denote the inverse function: \[ f^{-1}(x) = \frac{x - 4}{3} \] So the answer is \( \text{C) } f^{-1}(x) = \frac{x - 4}{3} \). Now for some extra fun facts: Finding function inverses is like finding a secret passage! Just as the original function transforms \( x \) to \( f(x) \), the inverse reverses the journey, giving us back \( x \) from \( f(x) \). This plays a crucial role in calculus, particularly when dealing with integrals and derivatives. In the real world, understanding inverses can help in various fields! For instance, in finance, if you know the future value of an investment based on the interest rate, you might want to find the initial investment. The formula for future value often has an inverse that allows you to determine the original amount, making it essential for budgeting and financial planning.