$1 \leftarrow \quad$ For $f(x)=\frac{5}{x-6}$ and $g(x)=\frac{7}{x}$, find the following composite functions and state the domain of each.
(a) $f \circ g$
(b) $g \circ f$
(c) $f \circ f$
(d) $g \circ g$
(a) $(f \circ g)(x)=\square($ Simplify your answer.)

Step 1 ：Given the functions \(f(x)=\frac{5}{x-6}\) and \(g(x)=\frac{7}{x}\), we are asked to find the composite function \(f \circ g(x)\) and its domain.

Step 2 ：The composite function \(f \circ g(x)\) means that we are substituting \(g(x)\) into \(f(x)\). So, we replace every \(x\) in \(f(x)\) with \(g(x)\).

Step 3 ：Substituting \(g(x)\) into \(f(x)\), we get \(f(g(x)) = \frac{5}{g(x)-6} = \frac{5}{\frac{7}{x}-6}\).

Step 4 ：Simplifying the above expression, we get \(f(g(x)) = \frac{-5x}{6x - 7}\).

Step 5 ：The domain of the composite function is the set of all real numbers except those that make the denominator zero in either \(f(x)\) or \(g(x)\). In this case, we need to exclude the values that make \(x-6=0\) in \(f(x)\) and \(x=0\) in \(g(x)\).

Step 6 ：Solving the equation \(6x - 7 = 0\) for \(x\), we get \(x = \frac{7}{6}\). So, we exclude \(\frac{7}{6}\) from the domain.

Step 7 ：Also, we exclude \(x=0\) from the domain because it makes the denominator of \(g(x)\) zero.

Step 8 ：Hence, the domain of the composite function \(f \circ g(x)\) is all real numbers except \(x = \frac{7}{6}\) and \(x = 0\).

Step 9 ：\(\boxed{Final Answer: (f \circ g)(x) = \frac{-5x}{6x - 7} \quad with \quad domain \quad x \in \mathbb{R} \setminus \{0, \frac{7}{6}\}}\)