what is the domain and range of f (x) = 3- x²

For the function \( f(x) = 3 - x^2 \), let's determine the domain and range.

### Domain:
The domain of a function consists of all the possible values of \( x \) that can be input into the function. Since \( f(x) = 3 - x^2 \) is a quadratic function (a polynomial), there are no restrictions on the values of \( x \). Therefore, the domain is:

\[
\text{Domain} = \mathbb{R} \quad \text{(all real numbers)}
\]

### Range:
The range of a function is the set of all possible output values. For \( f(x) = 3 - x^2 \), the function is a downward-opening parabola (since the coefficient of \( x^2 \) is negative). The vertex of the parabola gives the maximum value of the function.

The vertex of a quadratic function \( ax^2 + bx + c \) occurs at \( x = -\frac{b}{2a} \). In this case, \( a = -1 \), \( b = 0 \), and \( c = 3 \), so the vertex is at:

\[
x = -\frac{0}{2(-1)} = 0
\]

Substituting \( x = 0 \) back into the function:

\[
f(0) = 3 - (0)^2 = 3
\]

Since the parabola opens downward, the maximum value is 3, and the function can take any value less than or equal to 3. Therefore, the range is:

\[
\text{Range} = (-\infty, 3]
\]

### Summary:
- **Domain**: \( \mathbb{R} \) (all real numbers)
- **Range**: \( (-\infty, 3] \)

Would you like more details on how to determine the domain or range, or do you have any other questions?

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Here are 8 related questions to expand your understanding:
1. How do you determine the vertex of a quadratic function?
2. What is the significance of the sign of the coefficient of \( x^2 \) in determining the direction of a parabola?
3. How do you find the x-intercepts of \( f(x) = 3 - x^2 \)?
4. What are the general steps to graph a quadratic function?
5. How would the domain and range change if the function were \( f(x) = -3 - x^2 \)?
6. Can the function \( f(x) = 3 - x^2 \) have any maximum or minimum points other than the vertex?
7. What is the axis of symmetry for the function \( f(x) = 3 - x^2 \)?
8. How does the range of a function relate to its

Understand the domain and range of the quadratic function f(x) = 3 - x^2. Learn how to determine the domain (all real numbers) and range (values from negative infinity to 3) using vertex analysis. Suitable for high school students.

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