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The problem consists of three parts related to a function \( f(x) \) that satisfies a specific functional equation:

### Given Information:
\( f(2x) = a^{2 - 4x^2} \), where \( a > 0 \) and \( a \neq 1 \).

### Tasks:
1. **Find the explicit form of \( f(x) \)**.
2. **If \( a = 2 \)**, find the domain of the function \( y = \sqrt{f(x) - \frac{1}{4}} \)**.
3. **Discuss the domain of \( f(x) \)**.

---

### **1. Solving for \( f(x) \):**
We have the functional equation:
\[
f(2x) = a^{2 - 4x^2}
\]
Let’s perform a substitution for \( x = \frac{u}{2} \) so that \( f(u) = a^{2 - u^2} \). Therefore, the function \( f(x) \) is given by:
\[
f(x) = a^{2 - x^2}
\]
Thus, the general form of \( f(x) \) is:
\[
f(x) = a^{2 - x^2}
\]

### **2. Finding the Domain of \( y = \sqrt{f(x) - \frac{1}{4}} \) when \( a = 2 \):**

When \( a = 2 \), the function \( f(x) \) becomes:
\[
f(x) = 2^{2 - x^2}
\]
Now, we need to find the domain of the function:
\[
y = \sqrt{2^{2 - x^2} - \frac{1}{4}}
\]
For the square root to be defined, the expression inside must be non-negative:
\[
2^{2 - x^2} - \frac{1}{4} \geq 0
\]
This simplifies to:
\[
2^{2 - x^2} \geq \frac{1}{4}
\]
Taking the natural logarithm on both sides, we obtain:
\[
2 - x^2 \geq -2
\]
Solving for \( x \):
\[
x^2 \leq 4
\]
Thus, \( x \) must satisfy:
\[
-2 \leq x \leq 2
\]
So, the domain of \( y \) is \( x \in [-2, 2] \).

### **3. Discuss the Domain of \( f(x) \):**

The function \( f(x) = a^{2 - x^2} \), for \( a > 0 \), is always positive for all real \( x \), since any power of a positive number remains positive. Hence, the domain of \( f(x) \) is:
\[
\text{Domain of } f(x): (-\infty, \infty)
\]

---

Would you like more details or have any specific questions about these solutions?

Here are some related questions to consider:
1. What happens to \( f(x) \) as \( x \to \infty \)?
2. How does the value of \( a \) affect the shape of \( f(x) \)?
3. What is the range of the function \( f(x) \) for a specific value of \( a \)?
4. How would you find the inverse of the function \( f(x) \)?
5. Can the domain of \( y \) change if \( a \) is not equal to 2?

**Tip**: Always check the base \( a \) when dealing with exponential functions to understand how it affects the function’s growth or decay.

已知函数 f(x) 满足 f(2x) = a^{2 - 4x^2}，其中 a > 0 且 a ≠ 1。(1) 求 f(x) 的解析式；(2) 若 a=2，求函数 y = sqrt(f(x) - 1/4) 的定义域；(3) 讨论 f(x) 的值域。

Learn how to solve a function problem involving exponential functions and inequalities. The problem requires finding the explicit form of the function f(x), the domain of a new function based on f(x), and discussing the range of f(x). Suitable for students in grades 10-12.

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