The function written in your notebook appears to be:

$g(x) = \frac{x}{\sqrt{2x - 5}}$

This represents a rational function with a square root in the denominator.

To analyze the function:

1. Domain: The expression inside the square root must be non-negative (since we can't take the square root of a negative number in the reals). Thus, the domain is determined by solving:

$2x - 5 \geq 0$ $x \geq \frac{5}{2}$

So, the domain of $g(x)$ is $x \in \left[ \frac{5}{2}, \infty \right)$.

2. Range: Since the square root function outputs non-negative values and $x$ is in the numerator, the range of this function is determined by how the ratio behaves as $x$ increases.

3. Vertical Asymptote: There is no vertical asymptote because the denominator does not approach zero for any value of $x$ in the domain. However, as $x$ approaches $\frac{5}{2}$, the function tends to infinity.

Would you like more detailed information on how to handle asymptotes or range, or any further analysis?

Follow-up questions:

1. How do you determine the range of this function?
2. What happens to $g(x)$ as $x$ approaches infinity?
3. Can this function have horizontal asymptotes?
4. How does the square root in the denominator affect the behavior of $g(x)$?
5. How would the graph of this function look?

Tip: Always check the domain when square roots or other roots are involved in functions.