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.

Learn how to find the domain of the function f(x) = √(5x + 1) through a step-by-step explanation. This algebraic problem demonstrates how to solve inequalities to determine the range of valid x-values for square root functions. Suitable for students in Grades 9-11.

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