Evaluate 

*h*(8),

 where 

*h* = *g* ∘ *f*.

*f*(*x*) = 

3

*x*2 − 6

,    *g*(*x*) = 5*x*3 + 6

To evaluate \( h(8) \), where \( h = g \circ f \), we need to find the composite function \( h(x) = (g \circ f)(x) \).

### Step 1: Find the Composite Function \( h(x) = (g \circ f)(x) \)

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

The composite function \( h(x) = (g \circ f)(x) \) is defined as:
\[
h(x) = g(f(x))
\]

First, substitute \( f(x) = 3x^2 - 6 \) into \( g(x) \):

\[
h(x) = g(3x^2 - 6)
\]

Now, compute \( g(3x^2 - 6) \) by plugging in \( 3x^2 - 6 \) for \( x \) in the function \( g(x) \):

\[
g(3x^2 - 6) = 5(3x^2 - 6)^3 + 6
\]

### Step 2: Evaluate \( h(8) \)

To find \( h(8) \), substitute \( x = 8 \) into the expression for \( h(x) \):

\[
h(8) = g(f(8))


Learn how to evaluate h(8) where h = g ∘ f using composite functions g(x) = 5x^3 + 6 and f(x) = 3x^2 - 6. This problem explores the concept of composite functions in algebraic structures. Suitable for advanced high school students.

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