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 $x$.