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To solve this problem, we need to verify whether the two functions \( f(x) \) and \( g(x) \) are inverse functions of each other.

The functions given are:
- \( f(x) = \frac{1}{2} \sqrt{x - 4} \)
- \( g(x) = (2x)^2 + 4 \)

Two functions, \( f(x) \) and \( g(x) \), are inverses if:
- \( f(g(x)) = x \) and
- \( g(f(x)) = x \)

Additionally, we need to check the domains of both functions, because an inverse function should exist for all inputs in the domain of the original function and vice versa.

### Step 1: Check the domain of \( f(x) \)
The square root function \( \sqrt{x - 4} \) implies that \( x - 4 \geq 0 \), so the domain of \( f(x) \) is \( x \geq 4 \).

### Step 2: Check \( f(g(x)) \)
Substitute \( g(x) = (2x)^2 + 4 \) into \( f(x) \):
\[
f(g(x)) = \frac{1}{2} \sqrt{(2x)^2 + 4 - 4} = \frac{1}{2} \sqrt{(2x)^2} = \frac{1}{2} \cdot 2x = x
\]
Thus, \( f(g(x)) = x \), which is a necessary condition for inverse functions.

### Step 3: Check \( g(f(x)) \)
Substitute \( f(x) = \frac{1}{2} \sqrt{x - 4} \) into \( g(x) \):
\[
g(f(x)) = (2 \cdot \frac{1}{2} \sqrt{x - 4})^2 + 4 = (\sqrt{x - 4})^2 + 4 = (x - 4) + 4 = x
\]
Thus, \( g(f(x)) = x \), which is the other necessary condition for inverse functions.

### Step 4: Analyze the domain of \( g(x) \)
The function \( g(x) = (2x)^2 + 4 \) is defined for all real numbers \( x \).

### Conclusion:
While both \( f(g(x)) = x \) and \( g(f(x)) = x \), \( f(x) \) is only defined for \( x \geq 4 \). Therefore, the functions are not inverses over the entire domain of real numbers because the domain of \( f(x) \) is restricted.

The correct answer is **C**: No, because \( f \) is only defined on the domain \( x \geq 4 \).

Let me know if you need any further details, or have any questions!

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Here are 5 related questions:
1. What is the inverse of a general square root function?
2. How do you find the domain of a square root function?
3. Why is it important for functions to be defined on the same domain to be considered inverses?
4. Can a function with a restricted domain have an inverse on that restricted domain?
5. How do transformations like squaring and taking square roots affect the domain and range of a function?

**Tip:** When checking for inverse functions, always consider both the algebraic condition \( f(g(x)) = g(f(x)) = x \) and the domains of the functions involved.

Are f and g inverse functions on the domain of all real numbers? f(x) = \frac{1}{2} \sqrt{x - 4}, g(x) = (2x)^2 + 4

Are the functions f(x) = 1/2√(x-4) and g(x) = (2x)^2 + 4 inverses on the domain of all real numbers? This problem explores function composition, inverse functions, and domain restrictions. Suitable for high school students in Grades 10-12.

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