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To determine the domain of the given function \( f(x) \), we analyze each part:

The function \( f(x) \) is defined as:
\[
f(x) = 
\begin{cases} 
\frac{3}{7}x + 5 & \text{for } x < 7 \\
2 & \text{for } x \geq 7 
\end{cases}
\]

### Step 1: Analyzing the Domain
1. For \( x < 7 \), the expression \( \frac{3}{7}x + 5 \) is valid for all values of \( x \) less than 7. There are no restrictions here.
2. For \( x \geq 7 \), \( f(x) = 2 \) is valid for all values of \( x \geq 7 \).

### Combining the two parts
The function \( f(x) \) is therefore defined for all \( x \) such that \( x \in (-\infty, \infty) \).

### Domain in interval notation
The domain of the function in interval notation is:
\[
(-\infty, \infty)
\]

Would you like more details, or do you have any further questions?

### Here are 5 related questions:
1. What is the range of the given function \( f(x) \)?
2. How do we determine the continuity of this piecewise function at \( x = 7 \)?
3. What is the behavior of the function as \( x \to -\infty \)?
4. How would the graph of this piecewise function look?
5. What changes in the domain if \( f(x) = \frac{3}{7}x + 5 \) were restricted to \( x \leq 0 \) instead?

### Tip:
When working with piecewise functions, always consider each interval separately and then combine them to find the overall domain and range.

Find the domain and the range of the piecewise function: f(x) = (3/7)x + 5 for x < 7 and f(x) = 2 for x ≥ 7.

Learn how to find the domain and range of a piecewise function with a linear part (f(x) = (3/7)x + 5) and a constant part (f(x) = 2). This problem involves analyzing the behavior of the function for different intervals of x.

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