To find \((f \circ g)(x)\), which means \(f(g(x))\), we will substitute \(g(x)\) into \(f(x)\).
Given:
- \(f(x) = \sqrt{x - 4}\)
- \(g(x) = 5x - 6\)
Now we need to evaluate \(f(g(x))\):
- Substitute \(g(x)\) into \(f(x)\): \[ f(g(x)) = f(5x - 6) \]
- Now apply \(g(x)\) to the function \(f(x)\): \[ f(5x - 6) = \sqrt{(5x - 6) - 4} \]
- Simplify inside the square root: \[ f(5x - 6) = \sqrt{5x - 10} \]
Therefore, the answer for part (a) is: \[ \boxed{\sqrt{5x - 10}} \]
Now for part (b), we need to find \((g \circ f)(x)\), which means \(g(f(x))\).
Following a similar process:
- Substitute \(f(x)\) into \(g(x)\): \[ g(f(x)) = g(\sqrt{x - 4}) \]
- Now apply \(f(x)\) to the function \(g(x)\): \[ g(\sqrt{x - 4}) = 5(\sqrt{x - 4}) - 6 \]
Thus, the result for part (b) is: \[ \boxed{5\sqrt{x - 4} - 6} \]