The function you've given is: $f(x) = \sqrt{5x + 1}$

This function involves a square root, so its domain is limited to values of $x$ that keep the expression under the square root non-negative. Let's determine the domain:

Finding the Domain

The expression inside the square root must satisfy: $5x + 1 \geq 0$

Solving this inequality:

1. Subtract 1 from both sides: $5x \geq -1$
2. Divide by 5: $x \geq -\frac{1}{5}$

So, the domain of the function is: $x \geq -\frac{1}{5}$

Final Answer

The domain of the function $f(x) = \sqrt{5x + 1}$ is: $\left[ -\frac{1}{5}, \infty \right)$

Would you like a more detailed explanation or have any questions about this solution?

Here are five related questions to further your understanding:

1. What is the range of the function $f(x) = \sqrt{5x + 1}$?
2. How do you find the derivative of $f(x) = \sqrt{5x + 1}$?
3. What happens to the value of $f(x)$ as $x$ approaches infinity?
4. How would the domain change if the function were $f(x) = \sqrt{5x - 1}$?
5. Can you rewrite $f(x) = \sqrt{5x + 1}$ in exponential form?

Tip: Always check the domain of functions involving square roots or even roots, as they must be non-negative to ensure real values.