Given \( f(x)=5 \sqrt{x} \) and \( g(x)=5 x \), evaluate the following expressions. \( \begin{array}{llll}\text { (a) }(f \circ g)(4) & \text { (b) }(g \circ f)(2) & \text { (c) }(f \circ f)(1) & \text { (d) }(g \circ g)(0)\end{array} \) \( \begin{array}{l}\text { (a) }(f \circ g)(4)=\square \\ \text { (Type an exact answer, using radicals as needed. Simplify your answer.) }\end{array} \)

To evaluate the expressions, we will follow the definitions of function composition. The notation \( (f \circ g)(x) \) means \( f(g(x)) \). ### (a) \( (f \circ g)(4) \) 1. First, we need to evaluate \( g(4) \): \[ g(x) = 5x \implies g(4) = 5 \cdot 4 = 20 \] 2. Next, we evaluate \( f(g(4)) = f(20) \): \[ f(x) = 5\sqrt{x} \implies f(20) = 5\sqrt{20} \] 3. Simplifying \( \sqrt{20} \): \[ \sqrt{20} = \sqrt{4 \cdot 5} = 2\sqrt{5} \] Therefore, \[ f(20) = 5 \cdot 2\sqrt{5} = 10\sqrt{5} \] Thus, the answer for (a) is: \[ (f \circ g)(4) = 10\sqrt{5} \] ### (b) \( (g \circ f)(2) \) 1. First, we evaluate \( f(2) \): \[ f(x) = 5\sqrt{x} \implies f(2) = 5\sqrt{2} \] 2. Next, we evaluate \( g(f(2)) = g(5\sqrt{2}) \): \[ g(x) = 5x \implies g(5\sqrt{2}) = 5 \cdot (5\sqrt{2}) = 25\sqrt{2} \] Thus, the answer for (b) is: \[ (g \circ f)(2) = 25\sqrt{2} \] ### (c) \( (f \circ f)(1) \) 1. First, we evaluate \( f(1) \): \[ f(x) = 5\sqrt{x} \implies f(1) = 5\sqrt{1} = 5 \] 2. Next, we evaluate \( f(f(1)) = f(5) \): \[ f(5) = 5\sqrt{5} \] Thus, the answer for (c) is: \[ (f \circ f)(1) = 5\sqrt{5} \] ### (d) \( (g \circ g)(0) \) 1. First, we evaluate \( g(0) \): \[ g(x) = 5x \implies g(0) = 5 \cdot 0 = 0 \] 2. Next, we evaluate \( g(g(0)) = g(0) \): \[ g(0) = 0 \] Thus, the answer for (d) is: \[ (g \circ g)(0) = 0 \] ### Summary of Answers: - (a) \( (f \circ g)(4) = 10\sqrt{5} \) - (b) \( (g \circ f)(2) = 25\sqrt{2} \) - (c) \( (f \circ f)(1) = 5\sqrt{5} \) - (d) \( (g \circ g)(0) = 0 \)