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To solve the problem based on the given function \( f(x) = \frac{2 - x}{3} \) and the domain \( D(y) = \mathbb{R} \), let's analyze and solve the problem step by step.

### 1. Understanding the Function
The function is defined as:
\[
f(x) = \frac{2 - x}{3}
\]

### 2. Domain of the Function
The domain of \( f(x) \), denoted as \( D(x) \), includes all possible values of \( x \) for which \( f(x) \) is defined. In this case, \( f(x) \) is a linear function and is defined for all real numbers. Thus:
\[
D(x) = \mathbb{R}
\]

### 3. Range of the Function
Given that \( D(y) = \mathbb{R} \), we are asked to confirm that the range of the function \( f(x) \) also covers all real numbers.

To determine the range of \( f(x) \):
- Since the function is linear, it will produce all real numbers as \( x \) varies over all real numbers.
- This implies that for any \( y \in \mathbb{R} \), there exists an \( x \) such that \( y = f(x) = \frac{2 - x}{3} \).

Let's solve for \( x \) in terms of \( y \):
\[
y = \frac{2 - x}{3}
\]
Multiply both sides by 3:
\[
3y = 2 - x
\]
Now solve for \( x \):
\[
x = 2 - 3y
\]

Since \( y \) can take any value in \( \mathbb{R} \), \( x \) will also take any value in \( \mathbb{R} \). Therefore, the range of the function \( f(x) \) is \( \mathbb{R} \).

### Conclusion
- **Domain:** \( D(x) = \mathbb{R} \)
- **Range:** \( R(f) = \mathbb{R} \)

So, the given function \( f(x) = \frac{2 - x}{3} \) indeed has the range \( \mathbb{R} \), which matches the given condition \( D(y) = \mathbb{R} \).

Would you like further details on this solution, or do you have any questions?

Here are five additional related questions:
1. What is the effect of different coefficients on the linearity of \( f(x) = \frac{2 - x}{3} \)?
2. How do you find the inverse of the function \( f(x) = \frac{2 - x}{3} \)?
3. What happens to the range if the function is modified to \( f(x) = \frac{2 - x^2}{3} \)?
4. How can we graph the function \( f(x) = \frac{2 - x}{3} \) and its inverse?
5. How does changing the constant in the numerator affect the slope of the function?

**Tip:** When analyzing linear functions, always consider the slope (coefficient of \( x \)) and intercepts as key features for understanding the function's behavior and graph.

розвяжіть f(x) = \frac{2 - x}{3} with domain D(y) : R

Explore how to solve the function f(x) = (2 - x)/3 and determine the domain and range. This problem involves key algebraic concepts like functions, domain, and range. Suitable for high school students in Grades 9-12.

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