Find the domain of rational function. Show all steps

To find the domain of the rational function \( f(x) = \frac{2x^2 - 5x + 3}{x^2 + 5x} \), we need to determine the values of \( x \) for which the function is defined. A rational function is undefined where its denominator is zero. Therefore, we will follow these steps:

### Steps to Find the Domain

1. **Identify the denominator and set it equal to zero:**
   \[ x^2 + 5x = 0 \]

2. **Solve the equation \( x^2 + 5x = 0 \):**
   Factor the quadratic equation:
   \[ x(x + 5) = 0 \]

3. **Find the roots of the equation:**
   Set each factor equal to zero:
   \[ x = 0 \]
   \[ x = -5 \]

4. **Exclude the roots from the domain:**
   The values \( x = 0 \) and \( x = -5 \) make the denominator zero, so these values are not in the domain of the function.

### Conclusion
The domain of the function \( f(x) = \frac{2x^2 - 5x + 3}{x^2 + 5x} \) is all real numbers except \( x = 0 \) and \( x = -5 \). In interval notation, the domain is:

\[ (-\infty, -5) \cup (-5, 0) \cup (0, \infty) \]

### Summary of Steps
1. Set the denominator equal to zero: \( x^2 + 5x = 0 \).
2. Solve for \( x \): \( x = 0 \) and \( x = -5 \).
3. Exclude these \( x \)-values from the domain.

Would you like more details or have any questions?

Here are some more questions you might consider:
1. How to find the range of a rational function?
2. Can you explain the process of simplifying rational functions?
3. How do you find the intercepts of a rational function?
4. What is the process for finding holes in the graph of a rational function?
5. How can we analyze the end behavior of a rational function?

**Tip:** When finding the domain of a rational function, always focus on the denominator and determine where it equals zero, as these are the points that will be excluded from the domain.

Learn how to find the domain of a rational function step-by-step with an example. Understand where the function is undefined by analyzing its denominator. Suitable for students in Grades 10-12.

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