To analyze the function \( f(x) = 2 - 6x^2 \) over the interval \( -5 < x < 1 \), let’s examine a few key aspects:

### 1. Domain and Range:
- **Domain**: Given explicitly as \( -5 < x < 1 \).
- **Range**: To find the range, determine the maximum and minimum values of \( f(x) \) on the interval \( -5 < x < 1 \).

### 2. Behavior of \( f(x) \):
Since \( f(x) = 2 - 6x^2 \), this function is a downward-opening parabola (the coefficient of \( x^2 \) is negative), meaning it reaches a maximum at the vertex.

### 3. Vertex Calculation:
The vertex of a parabola \( ax^2 + bx + c \) is located at \( x = -\frac{b}{2a} \).
- Here, \( a = -6 \) and \( b = 0 \), so the vertex \( x \)-value is:
  \[
  x = -\frac{0}{2 \cdot (-6)} = 0
  \]
- Plugging \( x = 0 \) into \( f(x) \):
  \[
  f(0) = 2 - 6(0)^2 = 2
  \]
  So, \( f(x) \) has a maximum value of 2 at \( x = 0 \).

### 4. Boundary Values:
Evaluate \( f(x) \) at the boundaries of the domain:
- **At \( x = -5 \)**:
  \[
  f(-5) = 2 - 6(-5)^2 = 2 - 6 \cdot 25 = 2 - 150 = -148
  \]
- **At \( x = 1 \)**:
  \[
  f(1) = 2 - 6(1)^2 = 2 - 6 = -4
  \]

### 5. Range of \( f(x) \):
Given that \( f(x) \) reaches a maximum of 2 (at \( x = 0 \)) and ranges down to -148 at \( x = -5 \), the range of \( f(x) \) on \( -5 < x < 1 \) is:
\[
-148 < f(x) \leq 2
\]

### Summary
The function \( f(x) = 2 - 6x^2 \) over the interval \( -5 < x < 1 \) has:
- **Domain**: \( -5 < x < 1 \)
- **Range**: \( -148 < f(x) \leq 2 \)
  
Would you like further details on any step? 

### Related Questions:
1. How does the vertex form of a parabola affect its maximum and minimum values?
2. What would change in \( f(x) \) if the interval were \( -5 \leq x \leq 1 \)?
3. How does the negative coefficient in \( -6x^2 \) influence the shape of \( f(x) \)?
4. What is the process for finding critical points of \( f(x) \) on a closed interval?
5. How would you determine concavity for different functions like \( f(x) = 2 - 6x^2 \)?

#### Tip:
When analyzing the range of a function, always examine the function's critical points and endpoint behavior within the interval.

Analyze the function f(x) = 2 - 6x^2 over the interval -5 < x < 1

Discover how to analyze the quadratic function f(x) = 2 - 6x^2 within the interval -5 < x < 1. This solution provides a detailed explanation of finding the range and domain, locating the vertex, and evaluating boundary values for maximum and minimum values. Ideal for high school students exploring quadratic functions.

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