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The problem asks to find the domain and range of the quadratic function \( f(x) = -6x^2 + 5 \).

### Domain:
The domain of a quadratic function is the set of all possible input values (x-values) for which the function is defined. Since this is a polynomial function, it is defined for all real numbers. Thus, the domain is:
\[ (-\infty, \infty) \]

### Range:
To find the range, we need to analyze the behavior of the function.

The function is of the form \( f(x) = ax^2 + bx + c \), where \( a = -6 \), \( b = 0 \), and \( c = 5 \).

Since \( a = -6 \) is negative, the parabola opens downward, meaning the function has a maximum value at its vertex.

To find the vertex, we use the formula for the x-coordinate of the vertex:
\[
x = -\frac{b}{2a} = -\frac{0}{2(-6)} = 0
\]
At \( x = 0 \), we substitute into the function to find the maximum value:
\[
f(0) = -6(0)^2 + 5 = 5
\]
Thus, the maximum value of the function is 5. Since the parabola opens downward, the range of the function is all values less than or equal to 5. Therefore, the range is:
\[ (-\infty, 5] \]

### Final Answers:
- Domain: \( (-\infty, \infty) \)
- Range: \( (-\infty, 5] \)

Would you like more details or further clarification on any of the steps?

Here are 5 related questions to expand your understanding:
1. How would the domain and range change if the coefficient of \( x^2 \) were positive?
2. What is the significance of the vertex in determining the range of a quadratic function?
3. How can we determine whether a parabola opens upwards or downwards?
4. How would adding a linear term (like \( bx \)) affect the function’s domain and range?
5. How do you graph a quadratic function to visualize its domain and range?

**Tip:** Always check the coefficient of \( x^2 \) to determine whether the quadratic function has a maximum or minimum.

Find the domain and range of the function f(x) = -6x^2 + 5.

Understand how to determine the domain and range of the quadratic function f(x) = -6x^2 + 5. This problem covers essential concepts like identifying the vertex and analyzing the parabola's direction. Suitable for high school students studying quadratic functions.

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