To find the inverse function of \( f(x) = 3x + 4 \), we need to follow a systematic approach. Let's break it down step-by-step. ### Step 1: Understand the Function The function given is \( f(x) = 3x + 4 \). This is a linear function where: - The coefficient of \( x \) is 3 (this is the slope). - The constant term is 4 (this is the y-intercept). ### Step 2: Set Up the Equation To find the inverse, we start by replacing \( f(x) \) with \( y \): \[ y = 3x + 4 \] ### Step 3: Solve for \( x \) Next, we need to solve this equation for \( x \) in terms of \( y \): 1. Subtract 4 from both sides: \[ y - 4 = 3x \] 2. Now, divide both sides by 3: \[ x = \frac{y - 4}{3} \] ### Step 4: Write the Inverse Function Now that we have \( x \) in terms of \( y \), we can express the inverse function. We replace \( y \) with \( x \) to denote the inverse function: \[ f^{-1}(x) = \frac{x - 4}{3} \] ### Step 5: Identify the Correct Option Now, let's compare our derived inverse function \( f^{-1}(x) = \frac{x - 4}{3} \) with the options provided: - **A. \( \frac{x - 4}{3} \)** (This matches our derived inverse function) - B. \( \frac{x - 5}{5} \) - C. \( \frac{x + 3}{4} \) - D. \( \frac{x + 4}{3} \) The correct option is **A**. ### Step 6: Explain Why Other Options Are Incorrect - **B. \( \frac{x - 5}{5} \)**: This does not match our derived function. The subtraction and the divisor do not correspond to the original function's transformation. - **C. \( \frac{x + 3}{4} \)**: This option incorrectly adds 3 instead of subtracting 4, and it also has the wrong divisor. - **D. \( \frac{x + 4}{3} \)**: This option incorrectly adds 4 instead of subtracting it, and it also has the wrong divisor. ### Common Pitfalls - **Not isolating \( x \)**: A common mistake is to forget to isolate \( x \) correctly when solving for the inverse. - **Confusing addition and subtraction**: Be careful with signs; ensure you are correctly applying the operations as per the original function. ### Revision Summary - The inverse of a function \( f(x) \) is found by solving for \( x \) in terms of \( y \). - For \( f(x) = 3x + 4 \), the inverse is \( f^{-1}(x) = \frac{x - 4}{3} \). - Always check your derived inverse against the provided options to confirm correctness. - Be cautious with arithmetic operations to avoid common mistakes. By following these steps, you can confidently find the inverse of linear functions and understand the reasoning behind the correct answer.