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The problem asks to find the range of the function:

\[
y = \frac{2x+1}{x-4}
\]

### Steps to find the range:
1. **Identify the domain restrictions**: The function has a denominator \(x - 4\), so \(x \neq 4\) because division by zero is undefined.

2. **Express \(x\) in terms of \(y\)**:
   Solve for \(x\) by isolating it in terms of \(y\).
   \[
   y = \frac{2x+1}{x-4}
   \]
   Multiply both sides by \(x - 4\):
   \[
   y(x - 4) = 2x + 1
   \]
   Expand the left-hand side:
   \[
   yx - 4y = 2x + 1
   \]
   Bring all terms involving \(x\) to one side:
   \[
   yx - 2x = 4y + 1
   \]
   Factor out \(x\) on the left-hand side:
   \[
   x(y - 2) = 4y + 1
   \]
   Solve for \(x\):
   \[
   x = \frac{4y + 1}{y - 2}
   \]

3. **Analyze the new expression**:
   The new expression \(x = \frac{4y + 1}{y - 2}\) is undefined when \(y - 2 = 0\), i.e., \(y = 2\). Therefore, \(y = 2\) is excluded from the range.

### Conclusion:
The range of the function is all real numbers except \(y = 2\). In interval notation, the range is:

\[
\boxed{(-\infty, 2) \cup (2, \infty)}
\]

Would you like further clarification on this solution, or do you have any other questions?

### Related Questions:
1. What is the domain of the given function?
2. How does the asymptote at \(x = 4\) affect the graph of the function?
3. How would you sketch the graph of this rational function?
4. What is the horizontal asymptote of this function?
5. How would the function change if the numerator were \(3x + 2\) instead of \(2x + 1\)?

### Tip:
For rational functions, the values that make the denominator zero are points of discontinuity, often leading to vertical asymptotes. Always check for these when analyzing domain and range.

Find the range of the function y = (2x + 1) / (x - 4)

Learn how to find the range of the rational function y = (2x + 1) / (x - 4). This detailed solution includes domain restrictions, solving for x in terms of y, and understanding the function's range in interval notation. Suitable for Grades 10-12 students studying algebra and rational functions.

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