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.