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.