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We are tasked with finding the domain of the function:

\[
f(x) = \sqrt{4 - 5x}
\]

### Steps to determine the domain:
The square root function requires the expression under the square root to be non-negative because the square root of a negative number is not defined in the set of real numbers. Therefore, we need to solve the inequality:

\[
4 - 5x \geq 0
\]

1. Subtract 4 from both sides:

\[
-5x \geq -4
\]

2. Divide both sides by -5. Note that when dividing by a negative number, the inequality sign reverses:

\[
x \leq \frac{4}{5}
\]

### Conclusion:
The domain of \( f(x) \) is all \( x \) values such that \( x \leq \frac{4}{5} \).

In interval notation, the domain is:

\[
(-\infty, \frac{4}{5}]
\]

Let me know if you'd like a further breakdown of this process!

Here are five related questions to expand your understanding:
1. What is the range of \( f(x) \)?
2. How would the domain change if the function were \( f(x) = \sqrt{4 - 5x^2} \)?
3. How do you find the domain of a rational function?
4. Can the domain of \( f(x) = \sqrt{4 - 5x} \) be extended in the complex number system?
5. How does the concept of a domain apply to polynomial functions?

**Tip**: When solving inequalities involving negative coefficients, always remember to flip the inequality sign when dividing or multiplying by a negative number!

Find the domain of f and write your answer in interval notation.

f(x) = √(4 - 5x)

Learn how to find the domain of the function f(x) = √(4 - 5x). This problem covers important algebraic concepts, including inequalities and the properties of square roots. Step-by-step solution provided for students in Grades 9-11.

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