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We are given two functions and their domains:

1. \( f_1(x) = g(2x) \) with the natural domain \([-1, 3]\).
2. We are tasked with finding the natural domain of \( f_2(x) = g(\log_2{x}) \).

### Solution:

For the function \( f_1(x) = g(2x) \), we are told that the domain of \( g(2x) \) is \([-1, 3]\). This implies that the argument of the function, \( 2x \), lies within this interval:
\[
-1 \leq 2x \leq 3
\]
Solving for \( x \), we divide the entire inequality by 2:
\[
-\frac{1}{2} \leq x \leq \frac{3}{2}
\]
Thus, \( x \in \left[ -\frac{1}{2}, \frac{3}{2} \right] \).

Now, for the function \( f_2(x) = g(\log_2{x}) \), the argument \( \log_2{x} \) must also lie within the domain \([-1, 3]\). So, we have:
\[
-1 \leq \log_2{x} \leq 3
\]
To solve this inequality, we rewrite it as an exponential:
\[
2^{-1} \leq x \leq 2^3
\]
which simplifies to:
\[
\frac{1}{2} \leq x \leq 8
\]
Thus, the natural domain of \( f_2(x) \) is \( \left[ \frac{1}{2}, 8 \right] \).

### Final Answer:
The natural domain of \( f_2(x) = g(\log_2{x}) \) is \( \left[ \frac{1}{2}, 8 \right] \).

Would you like any additional details or have any questions?

Here are 5 related questions for further exploration:
1. How does changing the base of the logarithm affect the domain of a function like \( g(\log_b{x}) \)?
2. What is the domain of \( g(3x) \) if the domain of \( g(x) \) is \( [0, 5] \)?
3. How does the transformation of functions \( g(kx) \) affect the domain compared to \( g(x) \)?
4. What happens to the domain of \( f_2(x) = g(\ln{x}) \) compared to \( f_2(x) = g(\log_2{x}) \)?
5. How does shifting the function \( g(x) \) by a constant affect its domain?

**Tip:** When working with logarithmic functions, remember that the argument of the logarithm must always be positive.

The natural domain of the function f1(x) = g(2x) is [-1, 3]. What is the natural domain of f2(x) = g(log2(x))?

Learn how to find the domain of a logarithmic function f2(x) = g(log2(x)) given the domain of f1(x) = g(2x). This problem involves solving inequalities and applying properties of logarithms, suitable for high school students in grades 11-12 or early college level.

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