Find \( f(g(x)) \) and \( g(f(x)) \) and determine whether the pair of functions \( f \) and \( g \) are inverses of each othe \( f(x)=x \) and \( g(x)=x \) \( f(g(x))=x \) \( g(f(x))=\square \)

Let's analyze the given functions and their compositions to determine if \( f \) and \( g \) are inverses of each other. ### Given Functions: - \( f(x) = x \) - \( g(x) = x \) ### Composition of Functions: 1. **\( f(g(x)) \)**: \[ f(g(x)) = f(x) = x \] 2. **\( g(f(x)) \)**: \[ g(f(x)) = g(x) = x \] So, \[ g(f(x)) = x \] ### Determining Inverses: Two functions \( f \) and \( g \) are inverses of each other if: \[ f(g(x)) = x \quad \text{and} \quad g(f(x)) = x \quad \text{for all } x \text{ in their domains}. \] From our calculations: - \( f(g(x)) = x \) - \( g(f(x)) = x \) Since both compositions return \( x \), the functions \( f \) and \( g \) satisfy the condition for being inverse functions. ### Conclusion: Yes, the functions \( f(x) = x \) and \( g(x) = x \) are inverses of each other. \[ \boxed{ \begin{aligned} f(g(x)) &= x \\ g(f(x)) &= x \\ \text{Thus, } f \text{ and } g \text{ are inverses of each other.} \end{aligned} } \]