The given polynomial is:

$f(x) = -4x^2 + 9$

Domain:

For a polynomial function, the domain is always the set of all real numbers, because polynomials are defined for every real number. There are no restrictions like division by zero or taking square roots of negative numbers.

So, the domain of $f(x) = -4x^2 + 9$ is:

$\boxed{(-\infty, \infty)}$

Would you like more details on this, or is there anything else you'd like to explore about this polynomial?

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Related Questions:

1. What is the range of the function $f(x) = -4x^2 + 9$?
2. How do you find the vertex of the parabola for $f(x) = -4x^2 + 9$?
3. What is the axis of symmetry for the parabola $f(x) = -4x^2 + 9$?
4. How do you find the roots (x-intercepts) of the equation $f(x) = -4x^2 + 9$?
5. How does the negative coefficient in front of $x^2$ affect the graph of $f(x)$?

Tip:

When dealing with quadratic functions like this, remember that the domain is always all real numbers unless there are other constraints (like square roots or denominators).