To find the natural domain of the function $f(x) = \sqrt{2x + 3}$, we need to ensure that the expression inside the square root is non-negative, since the square root of a negative number is undefined in the real number system.

The condition for the square root to be defined is:

$2x + 3 \geq 0$

Step-by-step solution:

1. Solve the inequality: $2x + 3 \geq 0$

   Subtract 3 from both sides: $2x \geq -3$

   Divide by 2: $x \geq -\frac{3}{2}$

2. Conclusion: The natural domain of the function $f(x) = \sqrt{2x + 3}$ is all $x$-values such that $x \geq -\frac{3}{2}$.

In interval notation, the domain is: $\left[ -\frac{3}{2}, \infty \right)$

---

Would you like further details or have any questions?

Here are some related questions for practice:

1. What is the natural domain of $g(x) = \sqrt{5x - 4}$?
2. How would you find the domain of $h(x) = \frac{1}{\sqrt{x-2}}$?
3. What is the domain of $f(x) = \sqrt{x^2 - 4x + 3}$?
4. How does the domain change for $f(x) = \sqrt{3 - 2x}$?
5. What is the domain of $k(x) = \sqrt{x+5} + \sqrt{2x-1}$?

Tip: Always remember, for square root functions, the expression under the square root must be greater than or equal to zero to stay within the real number system!