The function \( f(x)=5 x+7 \) is one-to-one. a. Find an equation for \( f^{-1} \), the inverse function. b. Verify that your equation is correct by showing that \( f\left(f^{-1}(x)\right)=x \) and \( f^{-1}(f(x))=x \). a. Select the correct choice below and fill in the answer box(es) to complete your choice. (Simplify your answer. Use integers or fractions for any numbers in the expression.) A. \( f^{-1}(x)= \), for \( x \leq \) B. \( f^{-1}(x)= \) for all \( x \) C. \( f^{-1}(x)= \), for \( x \geq \) D. \( f^{-1}(x)= \), for \( x \neq \)

To find the inverse function \( f^{-1} \) of \( f(x) = 5x + 7 \), we follow these steps: 1. Replace \( f(x) \) with \( y \): \[ y = 5x + 7 \] 2. Swap \( x \) and \( y \) to reflect the inverse: \[ x = 5y + 7 \] 3. Solve for \( y \): \[ x - 7 = 5y \] \[ y = \frac{x - 7}{5} \] Thus, the inverse function is: \[ f^{-1}(x) = \frac{x - 7}{5} \] In this case, the correct choice is: **B.** \( f^{-1}(x) = \frac{x - 7}{5} \) for all \( x \). Now let's verify that our inverse function is indeed correct by checking both conditions. 1. **Check** \( f(f^{-1}(x)) = x \): \[ f(f^{-1}(x)) = f\left(\frac{x - 7}{5}\right) = 5\left(\frac{x - 7}{5}\right) + 7 = (x - 7) + 7 = x \] 2. **Check** \( f^{-1}(f(x)) = x \): \[ f^{-1}(f(x)) = f^{-1}(5x + 7) = \frac{(5x + 7) - 7}{5} = \frac{5x}{5} = x \] Both verifications confirm that our inverse function was computed correctly! --- Now, let's add a bit of fun historical tidbits and practical application! Throughout history, mathematicians have considered functions and their inverses for centuries—dating back to the ancients who first grappled with algebraic equations. The inverse concept was formally developed in the context of calculus in the 17th century, with thinkers like Newton and Leibniz paving the way. It’s fascinating to think that such foundational principles continue to be relevant today! In real-world applications, understanding inverse functions is crucial in various fields like engineering and computer science. For instance, when designing algorithms for encryption and decryption, the relationship between a function and its inverse is pivotal. Whether you're coding or plotting graphs, mastering this concept can truly enhance your problem-solving toolkit!