Question: Given $f(x) = 4x$ and $g(x) = 8x^2 + 5$, evaluate the following expressions.
(a) $(f \circ g)(4)$
(b) $(g \circ f)(2)$
(c) $(f \circ f)(1)$
(d) $(g \circ g)(0)$
1. **State the problem:** We are given two functions $f(x) = 4x$ and $g(x) = 8x^2 + 5$. We need to evaluate the compositions:
(a) $(f \circ g)(4)$ means $f(g(4))$
(b) $(g \circ f)(2)$ means $g(f(2))$
(c) $(f \circ f)(1)$ means $f(f(1))$
(d) $(g \circ g)(0)$ means $g(g(0))$
2. **Recall the composition rule:** For functions $f$ and $g$, $(f \circ g)(x) = f(g(x))$. This means we first evaluate $g(x)$, then plug that result into $f$.
3. **Calculate each part:**
(a) Calculate $g(4)$:
$$g(4) = 8 \times 4^2 + 5 = 8 \times 16 + 5 = 128 + 5 = 133$$
Then calculate $f(g(4)) = f(133)$:
$$f(133) = 4 \times 133 = 532$$
So, $(f \circ g)(4) = 532$.
(b) Calculate $f(2)$:
$$f(2) = 4 \times 2 = 8$$
Then calculate $g(f(2)) = g(8)$:
$$g(8) = 8 \times 8^2 + 5 = 8 \times 64 + 5 = 512 + 5 = 517$$
So, $(g \circ f)(2) = 517$.
(c) Calculate $f(1)$:
$$f(1) = 4 \times 1 = 4$$
Then calculate $f(f(1)) = f(4)$:
$$f(4) = 4 \times 4 = 16$$
So, $(f \circ f)(1) = 16$.
(d) Calculate $g(0)$:
$$g(0) = 8 \times 0^2 + 5 = 0 + 5 = 5$$
Then calculate $g(g(0)) = g(5)$:
$$g(5) = 8 \times 5^2 + 5 = 8 \times 25 + 5 = 200 + 5 = 205$$
So, $(g \circ g)(0) = 205$.
**Final answers:**
(a) 532
(b) 517
(c) 16
(d) 205