Question: Given f(x) = 2\sqrt{x} and g(x) = 7x, evaluate the following expressions.
(a) (f o g)(4) (b) (g o f)(2) (c) (f o f)(1) (d) (g o g)(0)
1. **State the problem:** We are given two functions $$f(x) = 2\sqrt{x}$$ and $$g(x) = 7x$$ and need to evaluate the compositions:
(a) $$(f \circ g)(4)$$
(b) $$(g \circ f)(2)$$
(c) $$(f \circ f)(1)$$
(d) $$(g \circ g)(0)$$
2. **Recall the composition of functions:**
$$(f \circ g)(x) = f(g(x))$$
$$(g \circ f)(x) = g(f(x))$$
3. **Evaluate each part step-by-step:**
**(a) Evaluate $$(f \circ g)(4) = f(g(4))$$**
- First find $$g(4) = 7 \times 4 = 28$$
- Then find $$f(28) = 2\sqrt{28} = 2 \times \sqrt{28}$$
- Simplify $$\sqrt{28} = \sqrt{4 \times 7} = 2\sqrt{7}$$
- So $$f(28) = 2 \times 2\sqrt{7} = 4\sqrt{7}$$
**Answer (a):** $$4\sqrt{7}$$
**(b) Evaluate $$(g \circ f)(2) = g(f(2))$$**
- First find $$f(2) = 2\sqrt{2}$$
- Then find $$g(2\sqrt{2}) = 7 \times 2\sqrt{2} = 14\sqrt{2}$$
**Answer (b):** $$14\sqrt{2}$$
**(c) Evaluate $$(f \circ f)(1) = f(f(1))$$**
- First find $$f(1) = 2\sqrt{1} = 2$$
- Then find $$f(2) = 2\sqrt{2}$$
**Answer (c):** $$2\sqrt{2}$$
**(d) Evaluate $$(g \circ g)(0) = g(g(0))$$**
- First find $$g(0) = 7 \times 0 = 0$$
- Then find $$g(0) = 0$$
**Answer (d):** $$0$$