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The image contains two questions regarding the domain and range of given functions. Let's address each question in detail:

1. **What is the domain and range of \( f(x) = 3x - x^2 \)?**

To determine the domain and range of the function \( f(x) = 3x - x^2 \), let's analyze it step-by-step.

### Domain of \( f(x) = 3x - x^2 \)
The domain of a function includes all the possible values of \( x \) for which the function is defined. Since \( f(x) = 3x - x^2 \) is a polynomial, it is defined for all real numbers \( x \). Thus, the domain is:
\[ \text{Domain} = (-\infty, \infty) \]

### Range of \( f(x) = 3x - x^2 \)
To find the range, we need to identify the possible values of \( f(x) \).

1. **Find the vertex of the parabola:**
   The function \( f(x) = 3x - x^2 \) is a downward-opening parabola (since the coefficient of \( x^2 \) is negative). The vertex form of a quadratic function \( ax^2 + bx + c \) has its vertex at \( x = -\frac{b}{2a} \).
   
   For \( f(x) = -x^2 + 3x \):
   \[ a = -1, \quad b = 3 \]
   \[ x = -\frac{3}{2(-1)} = \frac{3}{2} \]
   
2. **Find the maximum value of \( f(x) \):**
   Substitute \( x = \frac{3}{2} \) back into the function to find \( f \left( \frac{3}{2} \right) \):
   \[ f \left( \frac{3}{2} \right) = 3 \left( \frac{3}{2} \right) - \left( \frac{3}{2} \right)^2 = \frac{9}{2} - \frac{9}{4} = \frac{9}{2} - \frac{9}{4} = \frac{18}{4} - \frac{9}{4} = \frac{9}{4} = 2.25 \]

The maximum value of \( f(x) \) is 2.25, and since the parabola opens downwards, the range is:
\[ \text{Range} = (-\infty, 2.25] \]

2. **Find the domain and range of \( f(x) = 4 \) using its graph.**

### Domain of \( f(x) = 4 \)
The function \( f(x) = 4 \) is a constant function, meaning it is defined for all real numbers \( x \). Thus, the domain is:
\[ \text{Domain} = (-\infty, \infty) \]

### Range of \( f(x) = 4 \)
Since \( f(x) = 4 \) is a constant function, it only takes on the value 4, regardless of the input \( x \). Thus, the range is:
\[ \text{Range} = \{ 4 \} \]

Do you have any questions or need more details on any of these solutions?

Here are some related questions to further your understanding:

1. How do you determine the domain of a rational function?
2. What is the significance of the vertex in a quadratic function's graph?
3. How do you find the vertex of a quadratic function given in standard form?
4. Can the range of a quadratic function ever be all real numbers?
5. How would the function \( f(x) = 3x - x^2 + 5 \) differ in terms of domain and range?
6. What is the general process for finding the range of a function?
7. How do transformations (shifts, stretches) affect the domain and range of a function?
8. Can the range of a linear function be a single value?

**Tip:** For polynomial functions, the domain is always all real numbers unless specified otherwise by a context or additional constraints.

Explore detailed examples of finding the domain and range of functions, including quadratic and constant functions. Learn how to analyze functions like f(x) = 3x - x^2 and f(x) = 4 step-by-step, with explanations of their domains and ranges. Suitable for high school students and anyone interested in understanding function properties.

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