Evaluate
h(8),
where
h = g ∘ f.
f(x) =
3

x2 − 6
,    g(x) = 5x3 + 6

To evaluate \( h(8) \) where \( h = g \circ f \), we first need to understand what this notation means. The function \( h(x) \) is the composition of the functions \( g(x) \) and \( f(x) \), which means \( h(x) = g(f(x)) \).

Given:
- \( f(x) = \frac{3}{x^2} - 6 \)
- \( g(x) = 5x^3 + 6 \)

### Step 1: Find \( f(8) \)
First, we need to evaluate \( f(8) \):
\[
f(8) = \frac{3}{8^2} - 6
\]
\[
8^2 = 64
\]
\[
f(8) = \frac{3}{64} - 6 = \frac{3}{64} - \frac{384}{64} = \frac{3 - 384}{64} = \frac{-381}{64}
\]

### Step 2: Find \( g(f(8)) \)
Next, substitute \( f(8) = \frac{-381}{64} \) into \( g(x) \):
\[
g\left(\frac{-381}{64}\right) = 5\left(\frac{-381}{64}\right)^3 + 6
\]
Now, let's compute \( \left(\frac{-381}{64}\right)^3 \):
\[
\left(\frac{-381}{64}\right)^3 = \frac{(-381)^3}{64^3}
\]
\[
(-381)^3 = -381 \times -381 \times -381 = -55229241
\]
\[
64^3 = 262144
\]
Thus:
\[
g\left(\frac{-381}{64}\right) = 5 \times \frac{-55229241}{262144} + 6
\]
\[
= \frac{-276146205}{262144} + 6
\]
Convert 6 into a fraction with a denominator of 262144:
\[
6 = \frac{1572864}{262144}
\]
\[
g\left(\frac{-381}{64}\right) = \frac{-276146205 + 1572864}{262144} = \frac{-274573341}{262144}
\]

### Final Answer:
\[
h(8) = g(f(8)) = \frac{-274573341}{262144}
\]

This is the final evaluation of \( h(8) \).

Would you like further details or have any questions about the process? 

Here are 5 questions related to the topic:
1. What does it mean for a function to be the composition of two functions?
2. How do you simplify a fraction with large exponents?
3. Can you explain the impact of negative exponents on function evaluation?
4. What are the steps to convert a mixed number back into an improper fraction?
5. How would the process change if we had a different value for \( x \) in \( f(x) \)?

**Tip:** Always check your final answer by ensuring the intermediate steps follow the order of operations correctly, especially when dealing with negative exponents and large fractions.

Learn how to evaluate h(8) = g(f(8)) where h is the composition of functions g and f. Step-by-step solution provided for f(x) = 3/(x^2) - 6 and g(x) = 5x^3 + 6. Suitable for advanced high school students and college-level mathematics enthusiasts.

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