For the following function, a) determine whether it is one-to-one; \( b \) ) if it is one-to-one, find its inverse fun \( f(x)=x-6 \) Is the given function a one-to-one function? Select the correct choice below and, if necessary, fill in the as A. Yes, the function is one-to-one. The inverse function is \( \mathrm{f}^{-1}(\mathrm{x})= \) (Simplify your answer)

To determine whether the function \( f(x) = x - 6 \) is one-to-one, we can use the definition of a one-to-one function: a function is one-to-one if it never assigns the same value to two different domain elements. ### a) Is the function one-to-one? 1. **Check if \( f(a) = f(b) \) implies \( a = b \)**: \[ f(a) = a - 6 \] \[ f(b) = b - 6 \] If \( f(a) = f(b) \), then: \[ a - 6 = b - 6 \] Adding 6 to both sides gives: \[ a = b \] Since \( a = b \) whenever \( f(a) = f(b) \), the function is one-to-one. ### b) Find the inverse function To find the inverse function \( f^{-1}(x) \), we start by replacing \( f(x) \) with \( y \): \[ y = x - 6 \] Next, we solve for \( x \): \[ y + 6 = x \] Now, we express \( x \) in terms of \( y \): \[ x = y + 6 \] To write the inverse function, we switch \( x \) and \( y \): \[ f^{-1}(x) = x + 6 \] ### Conclusion A. Yes, the function is one-to-one. The inverse function is \( f^{-1}(x) = x + 6 \).