Let's solve each part of the composite functions given:

a. $f(g(-3))$

1. Compute $g(-3)$: $g(x) = x^2 + 5 \implies g(-3) = (-3)^2 + 5 = 9 + 5 = 14$
2. Now, compute $f(14)$: $f(x) = 4 - 2x \implies f(14) = 4 - 2 \times 14 = 4 - 28 = -24$ So, $f(g(-3)) = -24$.

b. $(g \circ h)(20)$

1. Compute $h(20)$: $h(x) = \frac{x^6}{7} \implies h(20) = \frac{20^6}{7}$ Let's calculate $20^6$ and then divide by 7. $20^6 = 64000000 \quad \text{so,} \quad h(20) = \frac{64000000}{7} \approx 9142857.14$
2. Now, compute $g(9142857.14)$: $g(x) = x^2 + 5 \implies g(9142857.14) = (9142857.14)^2 + 5$ Let's approximate the square and then add 5. $(9142857.14)^2 \approx 83571428567346.42 \quad \text{so,} \quad g(9142857.14) \approx 83571428567346.42 + 5$ $g(h(20)) \approx 83571428567351.42$.

c. $h(f(g(3)))$

1. Compute $g(3)$: $g(3) = 3^2 + 5 = 9 + 5 = 14$
2. Compute $f(14)$ (as previously calculated): $f(14) = -24$
3. Now, compute $h(-24)$: $h(x) = \frac{x^6}{7} \implies h(-24) = \frac{(-24)^6}{7}$ $(-24)^6 = 191102976 \quad \text{so,} \quad h(-24) = \frac{191102976}{7} \approx 27300425.14$ $h(f(g(3))) \approx 27300425.14$.

Would you like more details on any of these calculations?

Related Questions

1. How does the chain rule apply to composite functions like $f(g(x))$?
2. What would the result be if we were to use another value instead of -3 or 3 in the compositions?
3. How do the graphs of these functions look individually and as compositions?
4. What are the domains and ranges for these individual functions?
5. How would the results change if the functions $f, g, h$ were modified?

Tip

When dealing with composite functions, always evaluate the innermost function first and proceed outwards to understand the layering of function evaluations.