If \( f(x)=x+1 \) and \( g(x)=x-1 \) (a) \( f(g(x))=\square \) (b) \( g(f(x))=\square \) function of \( f(x) \) (c) Thus \( g(x) \) is called an \( \square \)

To solve the problems step by step: (a) We need to find \( f(g(x)) \). Given: - \( f(x) = x + 1 \) - \( g(x) = x - 1 \) First, we calculate \( g(x) \): \[ g(x) = x - 1 \] Now, we substitute \( g(x) \) into \( f(x) \): \[ f(g(x)) = f(x - 1) = (x - 1) + 1 = x \] So, \( f(g(x)) = x \). (b) Now we need to find \( g(f(x)) \). We already have: - \( f(x) = x + 1 \) Now, we substitute \( f(x) \) into \( g(x) \): \[ g(f(x)) = g(x + 1) = (x + 1) - 1 = x \] So, \( g(f(x)) = x \). (c) Since both \( f(g(x)) \) and \( g(f(x)) \) equal \( x \), we can conclude that \( g(x) \) is called an **inverse function** of \( f(x) \). Putting it all together: (a) \( f(g(x)) = x \) (b) \( g(f(x)) = x \) (c) Thus \( g(x) \) is called an **inverse function**.