The function given is $h(x) = \sqrt{4x - 5}$. To find the domain of this function, we need to determine the values of $x$ that make the expression under the square root non-negative.

Since the square root function is only defined for non-negative numbers, we require:

$4x - 5 \geq 0$

Step-by-step solution:

1. Solve for $x$: $4x - 5 \geq 0$ $4x \geq 5$ $x \geq \frac{5}{4}$

Thus, the domain of the function is all values of $x$ such that $x \geq \frac{5}{4}$.

Interval notation:

$\left[ \frac{5}{4}, \infty \right)$

Would you like more details or have any questions about this? Here are some additional related questions:

1. How would the domain change if it were $\sqrt{4x + 5}$?
2. What is the range of $h(x) = \sqrt{4x - 5}$?
3. How do you find the domain of a rational function?
4. What would the graph of this function look like?
5. How do square root functions affect the overall domain and range?

Tip: Always look for restrictions like square roots or divisions in functions when determining their domain.