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!

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.