Given the function \( f(x) = 2 - x^2 \), let's analyze some key aspects of it:

### 1. **Domain**:
The domain of \( f(x) \) includes all real numbers since there are no restrictions on the values of \( x \) (like division by zero or square roots of negative numbers). Therefore, the domain is:
\[
\text{Domain} = (-\infty, \infty)
\]

### 2. **Range**:
To find the range, observe that the function is a downward-opening parabola (since the coefficient of \( x^2 \) is negative). The maximum value occurs at the vertex of the parabola.

- The vertex is found at \( x = 0 \) because the general form \( ax^2 + bx + c \) simplifies to \( -x^2 + 2 \).
- Evaluating \( f(0) \):
  \[
  f(0) = 2 - 0^2 = 2
  \]
- Since the parabola opens downward, the range is:
  \[
  \text{Range} = (-\infty, 2]
  \]

### 3. **Intercepts**:
- **x-intercepts**: Set \( f(x) = 0 \).
  \[
  2 - x^2 = 0 \implies x^2 = 2 \implies x = \pm \sqrt{2}
  \]
  So, the x-intercepts are \( (\sqrt{2}, 0) \) and \( (-\sqrt{2}, 0) \).

- **y-intercept**: Set \( x = 0 \).
  \[
  f(0) = 2
  \]
  So, the y-intercept is \( (0, 2) \).

### 4. **Vertex**:
As noted, the vertex is at \( (0, 2) \).

### 5. **Graph Behavior**:
- The graph is symmetric about the y-axis because it's an even function (replacing \( x \) with \(-x\) gives the same result).
- As \( x \to \infty \) or \( x \to -\infty \), \( f(x) \to -\infty \).

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### Related Questions:
1. How do you find the maximum or minimum of a function using calculus?
2. What are the properties of even and odd functions?
3. How can you determine concavity and inflection points of a function?
4. What is the significance of intercepts in graph analysis?
5. How do transformations affect the graph of a parabola?

**Tip**: When analyzing quadratic functions, the sign of the coefficient of \( x^2 \) determines if the parabola opens upwards (positive) or downwards (negative).

Explore the detailed analysis of the quadratic function f(x) = 2 - x^2. This solution covers key topics like the domain, range, intercepts, and vertex of the function, along with a step-by-step breakdown of how to find them. Suitable for students in Grades 8-10.

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