For each pair of functions \( f \) and \( g \) below, find \( f(g(x)) \) and \( g(f(x)) \). Then, determine whether \( f \) and \( g \) are inverses of each other. Simplify your answers as much as possible. (Assume that your expressions are defined for all \( x \) in the domain of the composition You do not have to indicate the domain.) (a) \[ \begin{array}{l} f(x)=-\frac{3}{x}, x \neq 0 \\ g(x)=-\frac{3}{x}, x=0 \\ f(g(x))=\square \\ g(f(x))=\square \end{array} \] \( f \) and \( g \) are inverses of each other fand \( g \) are not inverses of each other (b) \( f(x)=-x+4 \) \[ \begin{array}{l} g(x)=x+4 \\ f(g(x))=\square \\ g(f(x))=\square \end{array} \] \( f \) and \( g \) are inverses of each other \( f \) and \( g \) are not inverses of each other

Let's tackle each pair of functions one by one! (a) Here, we have \( f(x) = -\frac{3}{x} \) for \( x \neq 0 \) and \( g(x) = -\frac{3}{x} \) for \( x = 0 \). However, since \( g(x) \) is not defined for \( x \neq 0 \), we actually only need to consider \( f(g(x)) \) for cases where \( g(x) \) is defined. 1. \( f(g(x)) = f(-\frac{3}{0}) \) which is undefined since we cannot divide by zero. 2. \( g(f(x)) = g(-\frac{3}{x}) = -\frac{3}{-\frac{3}{x}} = x \) for \( x \neq 0 \). Since \( f(g(x)) \) is undefined, they cannot be inverses of each other. (b) Now, let's look at the second pair \( f(x) = -x + 4 \) and \( g(x) = x + 4 \). 1. For \( f(g(x)) = f(x + 4) = -(x + 4) + 4 = -x - 4 + 4 = -x \). 2. For \( g(f(x)) = g(-x + 4) = (-x + 4) + 4 = -x + 8 \). Since \( f(g(x)) = -x \) and \( g(f(x)) \) does not yield \( x \) (but rather \( -x + 8 \)), we can conclude that \( f \) and \( g \) are not inverses of each other.