The function \( Y = \sqrt{3x + 1} \) represents a square root function. Here's how we interpret and analyze it:

### 1. **Domain**:
To determine the domain of \( Y \), we need to make sure that the expression inside the square root is non-negative (because the square root of a negative number is not a real number).

\[
3x + 1 \geq 0
\]

\[
3x \geq -1
\]

\[
x \geq -\frac{1}{3}
\]

So, the domain of \( Y \) is:

\[
x \in \left[-\frac{1}{3}, \infty\right)
\]

### 2. **Range**:
Since the square root function always yields non-negative values, the range of \( Y \) will be:

\[
Y \geq 0
\]

In interval notation, the range is:

\[
Y \in [0, \infty)
\]

### 3. **Intercepts**:
   - **Y-intercept**: Set \( x = 0 \):
     \[
     Y = \sqrt{3(0) + 1} = \sqrt{1} = 1
     \]
     The Y-intercept is \( (0, 1) \).
   - **X-intercept**: Set \( Y = 0 \):
     \[
     \sqrt{3x + 1} = 0 \implies 3x + 1 = 0 \implies x = -\frac{1}{3}
     \]
     The X-intercept is \( \left(-\frac{1}{3}, 0\right) \).

### 4. **Behavior**:
   - As \( x \to \infty \), \( Y \to \infty \).
   - The function is increasing since the square root of a larger number yields a larger result.

### 5. **Graph**:
   - The graph starts at \( \left(-\frac{1}{3}, 0\right) \) and increases steadily, moving upwards and to the right.

Would you like further details or have any questions about this function?

### Related Questions:
1. How would the domain and range change if the function were \( Y = \sqrt{3x - 1} \)?
2. What would happen if a negative sign were placed in front of the square root, like \( Y = -\sqrt{3x + 1} \)?
3. How can you determine the domain of a function involving square roots and fractions?
4. What are the transformations applied if the function were \( Y = \sqrt{3x + 1} + 2 \)?
5. How do square root functions differ from quadratic functions in terms of shape and behavior?

### Tip:
To find the domain of a square root function, always set the expression under the square root \( \geq 0 \) and solve for \( x \).

Explore the square root function Y = √(3x + 1) with a detailed breakdown of its domain, range, intercepts, and behavior. Learn how to analyze and graph square root functions, suitable for students in Grades 9-11.

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