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Let's solve each of these functions by identifying the **vertical asymptotes**, **horizontal asymptotes**, **domain**, and **range**.

### a) \( f(x) = \frac{-4}{x} \)

1. **Vertical Asymptote**: The vertical asymptote occurs when the denominator is zero. For \( f(x) \), the denominator is \( x \), so the vertical asymptote is at:
   \[
   x = 0
   \]

2. **Horizontal Asymptote**: To find the horizontal asymptote, analyze the behavior as \( x \to \infty \) or \( x \to -\infty \). Since this is a rational function where the degree of the numerator is equal to the degree of the denominator:
   \[
   \text{Horizontal Asymptote: } y = 0
   \]

3. **Domain**: The domain excludes points where the denominator is zero. Therefore, the domain is:
   \[
   x \in (-\infty, 0) \cup (0, \infty)
   \]

4. **Range**: Since \( f(x) \) can take any real value except 0 (as the function never equals 0):
   \[
   \text{Range: } y \in (-\infty, 0) \cup (0, \infty)
   \]

---

### b) \( f(x) = \frac{4}{x-1} + 1 \)

1. **Vertical Asymptote**: Set the denominator equal to zero:
   \[
   x - 1 = 0 \quad \Rightarrow \quad x = 1
   \]
   So the vertical asymptote is at \( x = 1 \).

2. **Horizontal Asymptote**: The degree of the numerator and denominator is the same, so the horizontal asymptote is determined by the ratio of the leading coefficients:
   \[
   \frac{4}{1} = 4 \quad \text{and adding 1:} \quad y = 1
   \]
   So the horizontal asymptote is \( y = 1 \).

3. **Domain**: The domain excludes the value \( x = 1 \):
   \[
   x \in (-\infty, 1) \cup (1, \infty)
   \]

4. **Range**: Since the function can take values arbitrarily close to 1 but never equal to 1, the range is:
   \[
   y \in (1, \infty) \quad \text{(for negative \( f(x) \) values, the range also includes negative \( y \)-values.)}
   \]

---

### c) \( f(x) = \frac{-3}{x-1} - 1 \)

1. **Vertical Asymptote**: Again, set the denominator equal to zero:
   \[
   x - 1 = 0 \quad \Rightarrow \quad x = 1
   \]
   So the vertical asymptote is \( x = 1 \).

2. **Horizontal Asymptote**: The degree of the numerator and denominator is the same, so the horizontal asymptote is given by the ratio of the leading coefficients:
   \[
   \frac{-3}{1} = -3 \quad \text{and subtracting 1:} \quad y = -1
   \]
   So the horizontal asymptote is \( y = -1 \).

3. **Domain**: The domain excludes \( x = 1 \):
   \[
   x \in (-\infty, 1) \cup (1, \infty)
   \]

4. **Range**: The range of the function is all real numbers, except for \( y = -1 \), since the function cannot cross the horizontal asymptote:
   \[
   y \in (-\infty, -1) \cup (-1, \infty)
   \]

---

### d) \( f(x) = \frac{-3}{x} \)

1. **Vertical Asymptote**: The denominator is zero when \( x = 0 \):
   \[
   x = 0
   \]
   So the vertical asymptote is \( x = 0 \).

2. **Horizontal Asymptote**: Since the degrees of the numerator and denominator are the same, the horizontal asymptote is:
   \[
   y = 0
   \]

3. **Domain**: The domain excludes \( x = 0 \

Identify the vertical asymptotes, horizontal asymptotes, domain, and range of each of the following functions: a) f(x) = -4/x, b) f(x) = 4/(x-1) + 1, c) f(x) = -3/(x-1) - 1, d) f(x) = -3/x.

Learn how to identify vertical and horizontal asymptotes, domain, and range for the rational functions f(x) = -4/x, f(x) = 4/(x-1) + 1, f(x) = -3/(x-1) - 1, and f(x) = -3/x. This guide covers key concepts in rational functions and their properties, suitable for Grades 10-12.

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