Find the domain and range of the function f(x)=-4x^2-5
.

The domain is
(Enter your answer in interval notation)

The range is
(Enter your answer in interval notation)

To find the domain and range of the function \( f(x) = -4x^2 - 5 \), let's analyze the function step by step.

### Domain

The domain of a function is the set of all possible input values (x-values) that the function can accept. For the function \( f(x) = -4x^2 - 5 \):

- The expression \( -4x^2 - 5 \) is defined for all real numbers \( x \).
- There are no restrictions on \( x \) since there are no denominators, square roots, logarithms, or any other expressions that would limit the values of \( x \).

Therefore, the domain of \( f(x) \) is all real numbers.

**Domain:** \( (-\infty, \infty) \)

### Range

The range of a function is the set of all possible output values (y-values) that the function can produce. For the function \( f(x) = -4x^2 - 5 \):

- \( f(x) = -4x^2 - 5 \) is a downward-opening parabola because the coefficient of \( x^2 \) is negative (\( -4 \)).
- The vertex of this parabola will give us the maximum value of \( f(x) \).

To find the vertex:
- The standard form of a parabola is \( ax^2 + bx + c \). In this case, \( a = -4 \), \( b = 0 \), and \( c = -5 \).
- The x-coordinate of the vertex is given by \( x = -\frac{b}{2a} \). Here, \( b = 0 \), so \( x = 0 \).

Substituting \( x = 0 \) into the function to find the y-coordinate of the vertex:
\[ f(0) = -4(0)^2 - 5 = -5 \]

The maximum value of \( f(x) \) is -5. Since the parabola opens downward, \( f(x) \) takes on all values less than or equal to -5.

**Range:** \( (-\infty, -5] \)

### Summary

**Domain:** \( (-\infty, \infty) \)

**Range:** \( (-\infty, -5] \)

Would you like any further details or have any questions?

Here are 5 related questions you might find useful:

1. How do you find the vertex of a quadratic function?
2. How do you determine if a parabola opens upward or downward?
3. What are the steps to convert a quadratic function from standard form to vertex form?
4. Can you explain how to find the domain of a function involving square roots?
5. How do you solve a quadratic equation using the quadratic formula?

**Tip:** Remember that the vertex of a parabola in standard form \( ax^2 + bx + c \) can be found using \( x = -\frac{b}{2a} \). This helps determine the maximum or minimum value of the quadratic function.

Explore how to determine the domain and range of the quadratic function f(x) = -4x^2 - 5. This detailed solution explains the concepts of quadratic functions, including finding the vertex to determine the range and understanding the domain. Ideal for high school students.

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