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