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The problem asks for the **domain** and **range** of the quadratic function \( f(x) = -4x^2 + 3x - 3 \).

### Domain
Since this is a quadratic function, it is defined for all real values of \( x \). Therefore, the domain is:
\[
\boxed{ \mathbb{R} } \text{ (the set of all real numbers)}.
\]

### Range
The range of a quadratic function depends on whether it opens upward or downward. In this case, the coefficient of \( x^2 \) is \( -4 \), which is negative, so the parabola opens downward. To find the range, we need to determine the **maximum value** of the function.

#### Step 1: Find the vertex
The vertex of a parabola given by \( f(x) = ax^2 + bx + c \) can be found using the formula for the x-coordinate of the vertex:
\[
x = \frac{-b}{2a}.
\]
Here, \( a = -4 \) and \( b = 3 \), so:
\[
x = \frac{-3}{2(-4)} = \frac{3}{8}.
\]

#### Step 2: Calculate the maximum value
Now substitute \( x = \frac{3}{8} \) into the function to find the corresponding \( y \)-value:
\[
f\left( \frac{3}{8} \right) = -4\left( \frac{3}{8} \right)^2 + 3\left( \frac{3}{8} \right) - 3.
\]
First, calculate \( \left( \frac{3}{8} \right)^2 = \frac{9}{64} \):
\[
f\left( \frac{3}{8} \right) = -4 \times \frac{9}{64} + 3 \times \frac{3}{8} - 3 = -\frac{36}{64} + \frac{9}{8} - 3.
\]
Simplify:
\[
f\left( \frac{3}{8} \right) = -\frac{9}{16} + \frac{9}{8} - 3 = -0.5625 + 1.125 - 3 = -2.4375.
\]

So, the maximum value of the function is approximately \( -2.4375 \).

#### Step 3: Range
Since the parabola opens downward, the range is:
\[
\boxed{ (-\infty, -2.4375] }.
\]

Let me know if you'd like more details or clarifications!

### Related Questions:
1. What is the general formula for the vertex of a quadratic function?
2. How do you determine if a quadratic function opens upward or downward?
3. How do you find the domain of a polynomial function?
4. What role does the discriminant play in the graph of a quadratic function?
5. Can a quadratic function have both a maximum and a minimum point?

### Tip:
Remember that for any quadratic function \( ax^2 + bx + c \), if \( a > 0 \), the parabola opens upward and has a minimum, while if \( a < 0 \), it opens downward and has a maximum.

Find the range and domain of the function f(x) = -4x^2 + 3x - 3.

Learn how to find the domain and range of the quadratic function f(x) = -4x^2 + 3x - 3. This tutorial covers important concepts like quadratic functions, vertex formula, and the behavior of parabolas. Suitable for College Algebra students.

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