The function \( f(x)=x+14 \) is one-to-one. a. Find an equation for \( f^{-1}(x) \), 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. \( f^{-1}(x)=\square \), for \( x \leq \square \) B. \( f^{-1}(x)=\square \), for \( x \neq \square \) C. \( f^{-1}(x)=\square \), for \( x \geq \square \) D. \( f^{-1}(x)=\square \), for all \( x \)

To find the inverse function \( f^{-1}(x) \) for \( f(x) = x + 14 \), we start by setting \( y = f(x) \), which gives us \( y = x + 14 \). To find the inverse, we solve for \( x \) in terms of \( y \): 1. Rearranging the equation: \( x = y - 14 \). 2. Thus, \( f^{-1}(x) = x - 14 \). So, we can select option D and fill in the blanks: **D. \( f^{-1}(x)=x-14 \), for all \( x \)**. Next, let's verify that our equation is correct: 1. Compute \( f(f^{-1}(x)) \): \[ f(f^{-1}(x)) = f(x - 14) = (x - 14) + 14 = x. \] 2. Now compute \( f^{-1}(f(x)) \): \[ f^{-1}(f(x)) = f^{-1}(x + 14) = (x + 14) - 14 = x. \] Since both verifications hold true, our inverse function is confirmed to be correct!