The given graph appears to be a rational function with vertical and horizontal asymptotes. The blue curve represents the function, while the red dashed lines indicate the asymptotes.

Observations:

• There is a vertical asymptote at $x = 2$, meaning the function is undefined at this point.
• There is a horizontal asymptote at $y = 3$, meaning the function approaches $y = 3$ but never reaches it.

Range Analysis:

• The function takes all values except $y = 3$.
• As $x \to -\infty$ or $x \to \infty$, the function approaches $y = 3$.
• The function decreases toward negative infinity near $x = 2$ and increases toward positive infinity on the other side of $x = 2$.

Thus, the range of the function is: $(-\infty, 3) \cup (3, \infty)$ which means all real numbers except $y = 3$.

Would you like a detailed step-by-step explanation of how to determine the range?

Related Questions:

1. How do vertical asymptotes affect the domain of a function?
2. What does a horizontal asymptote tell us about a function's end behavior?
3. How can you algebraically determine the range of a rational function?
4. What transformations could shift the asymptotes of this function?
5. How would the range change if the function had no horizontal asymptote?

Tip:

When analyzing rational functions, always look for asymptotes to determine restrictions in the range and domain.