https://file.aiquickdraw.com/mathbot/file/1725548650898veuxynwc.jpg

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.

Explore the rational function g(x) = x / sqrt(2x - 5), including domain restrictions, behavior at infinity, and how square roots in the denominator affect the function. This detailed analysis is ideal for high school students studying algebra and functions.

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