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!