Let's analyze the function:

$f(x) = \frac{2x^2 - 5x + 3}{x^2 + 5x}$

(i) Finding the Horizontal and Vertical Asymptotes

1. Horizontal Asymptotes:

To find the horizontal asymptotes, we consider the degrees of the numerator and the denominator. The degrees of both the numerator and the denominator are the same (degree 2), so the horizontal asymptote is determined by the ratio of the leading coefficients of the highest degree terms.

• Numerator (highest degree term): $2x^2$
• Denominator (highest degree term): $x^2$

The horizontal asymptote is:

$y = \frac{\text{Leading coefficient of numerator}}{\text{Leading coefficient of denominator}} = \frac{2}{1} = 2$

Thus, the horizontal asymptote is:

$y = 2$

2. Vertical Asymptotes:

Vertical asymptotes occur when the denominator equals zero, provided the numerator does not also equal zero at those points (which would indicate a hole instead of an asymptote). To find the vertical asymptotes, we solve:

$x^2 + 5x = 0$

Factor this equation:

$x(x + 5) = 0$

So, the solutions are:

$x = 0 \quad \text{or} \quad x = -5$

Thus, the vertical asymptotes are:

$x = 0 \quad \text{and} \quad x = -5$

(ii) Finding the Domain of the Function

The domain of a rational function is all real numbers except where the denominator is zero. From the vertical asymptote calculation, we know the denominator is zero when:

$x(x + 5) = 0 \quad \Rightarrow \quad x = 0 \quad \text{or} \quad x = -5$

Thus, the function is undefined at $x = 0$ and $x = -5$. Therefore, the domain is:

$\boxed{(-\infty, -5) \cup (-5, 0) \cup (0, \infty)}$

Summary:

• Horizontal asymptote: $y = 2$
• Vertical asymptotes: $x = 0$ and $x = -5$
• Domain: $(- \infty, -5) \cup (-5, 0) \cup (0, \infty)$

Would you like further explanation on any step, or additional details?

Here are 5 related questions to consider:

1. How do you determine horizontal asymptotes for different degrees in the numerator and denominator?
2. What happens if both the numerator and denominator have a common factor?
3. How do we find slant (oblique) asymptotes for rational functions?
4. Can a function have more than one horizontal asymptote?
5. How would you graph this function, showing asymptotes and key points?

Tip: Always check for holes in the graph by factoring both the numerator and denominator—cancelled factors indicate holes, not vertical asymptotes.