Given the functions $f(x) = 4x - 2$ and $h(x) = \frac{x}{2} + 1$, we need to find $h(f(x))$ and then solve the equation $h(f(x)) = 12$.

Step 1: Find $h(f(x))$

To find $h(f(x))$, substitute $f(x)$ into $h(x)$:

$h(f(x)) = h(4x - 2)$

Since $h(x) = \frac{x}{2} + 1$, substitute $4x - 2$ for $x$:

$h(f(x)) = \frac{4x - 2}{2} + 1$

Simplify:

$h(f(x)) = \frac{4x}{2} - \frac{2}{2} + 1 = 2x - 1 + 1 = 2x$

So, $h(f(x)) = 2x$.

Step 2: Solve the equation $h(f(x)) = 12$

We know $h(f(x)) = 2x$. Now, set this equal to 12:

$2x = 12$

Solve for $x$:

$x = \frac{12}{2} = 6$

Final Answer:

The value of $x$ that satisfies $h(f(x)) = 12$ is $x = 6$.

Would you like more details, or do you have any questions? Here are some related questions to consider:

1. How would you solve for $x$ if the functions were different?
2. What is the significance of composite functions in mathematics?
3. Can you explain the process of function composition in more depth?
4. What if the equation was $h(f(x)) = 0$? How would that change the solution?
5. How do you find the inverse of the function $h(x)$?

Tip: When composing functions, always start by substituting the inner function into the outer function and simplify before solving any resulting equations.