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(i) Find the horizontal and vertical asymptotes. 

(ii) Find the domain of rational function. Show all steps.

Let's analyze the given rational function \( f(x) = \frac{2x^2 - 5x + 3}{x^2 + 5x} \) to find its horizontal and vertical asymptotes, and determine its domain.

### (i) Finding the Horizontal and Vertical Asymptotes

**Horizontal Asymptote:**

To find the horizontal asymptote, we need to compare the degrees of the polynomial in the numerator and the polynomial in the denominator.

- The degree of the numerator \( 2x^2 - 5x + 3 \) is 2.
- The degree of the denominator \( x^2 + 5x \) is 2.

Since the degrees are equal, the horizontal asymptote is determined by the ratio of the leading coefficients of the numerator and the denominator.

\[ f(x) = \frac{2x^2 - 5x + 3}{x^2 + 5x} \]
The leading coefficient of the numerator is 2, and the leading coefficient of the denominator is 1.

Therefore, the horizontal asymptote is:
\[ y = \frac{2}{1} = 2 \]

**Vertical Asymptote:**

Vertical asymptotes occur where the denominator is zero, provided the numerator is not zero at those points.

Set the denominator equal to zero and solve for \( x \):

\[ x^2 + 5x = 0 \]
\[ x(x + 5) = 0 \]

So, \( x = 0 \) and \( x = -5 \).

We need to check if these values also make the numerator zero. 

For \( x = 0 \):
\[ 2(0)^2 - 5(0) + 3 = 3 \]
which is not zero.

For \( x = -5 \):
\[ 2(-5)^2 - 5(-5) + 3 = 2(25) + 25 + 3 = 50 + 25 + 3 = 78 \]
which is not zero.

Thus, the vertical asymptotes are:
\[ x = 0 \]
\[ x = -5 \]

### (ii) Finding the Domain

The domain of a rational function is all real numbers except where the denominator is zero.

The denominator \( x^2 + 5x \) is zero when:
\[ x(x + 5) = 0 \]
\[ x = 0 \]
\[ x = -5 \]

Therefore, the domain is all real numbers except \( x = 0 \) and \( x = -5 \).

### Summary:
1. **Horizontal Asymptote:** \( y = 2 \)
2. **Vertical Asymptotes:** \( x = 0 \) and \( x = -5 \)
3. **Domain:** \( x \in \mathbb{R} \setminus \{0, -5\} \)

Would you like any further details or explanations on any of the steps?

Here are some related questions you might be interested in:
1. How do you find the x-intercepts and y-intercepts of a rational function?
2. How do you simplify complex rational expressions?
3. How do you determine slant (oblique) asymptotes for rational functions?
4. What is the process for graphing rational functions?
5. How do you find the inverse of a rational function?

**Tip:** Always check for common factors between the numerator and denominator before determining vertical asymptotes to simplify the rational function if possible.

Learn how to find horizontal and vertical asymptotes, and determine the domain of rational functions with a detailed step-by-step solution. This problem covers key concepts in rational functions, including asymptotes and domain restrictions. Suitable for students in Grades 11-12.

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