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To solve the problem, we need to analyze how the graph of \( f(x) = \frac{1}{x+4} \) can be obtained from the graph of \( y = \frac{1}{x} \), and also to discuss the domain, range, and intervals of increase and decrease.

### Step-by-Step Solution:

1. **Transformation:**
   - The given function is \( f(x) = \frac{1}{x+4} \).
   - This is a horizontal shift of the basic function \( y = \frac{1}{x} \).
   - To transform \( y = \frac{1}{x} \) into \( f(x) = \frac{1}{x+4} \), we shift the graph 4 units to the left.
   - Therefore, to obtain the graph of \( f \), shift the graph of \( y = \frac{1}{x} \) 4 units to the left.

2. **Graphing:**
   - The basic graph of \( y = \frac{1}{x} \) has vertical asymptote at \( x = 0 \) and horizontal asymptote at \( y = 0 \).
   - After shifting 4 units to the left, the vertical asymptote of \( f(x) = \frac{1}{x+4} \) will be at \( x = -4 \), and the horizontal asymptote will remain at \( y = 0 \).

3. **Domain:**
   - The domain of \( y = \frac{1}{x} \) is all real numbers except \( x = 0 \).
   - For \( f(x) = \frac{1}{x+4} \), the domain is all real numbers except \( x = -4 \).
   - Hence, the domain of \( f \) is \( (-\infty, -4) \cup (-4, \infty) \).

4. **Range:**
   - The range of \( y = \frac{1}{x} \) is all real numbers except \( y = 0 \).
   - Similarly, for \( f(x) = \frac{1}{x+4} \), the range remains all real numbers except \( y = 0 \).
   - Hence, the range of \( f \) is \( (-\infty, 0) \cup (0, \infty) \).

5. **Intervals of Increase and Decrease:**
   - The function \( y = \frac{1}{x} \) decreases on \( (-\infty, 0) \) and increases on \( (0, \infty) \).
   - After the horizontal shift, \( f(x) = \frac{1}{x+4} \) will decrease on \( (-\infty, -4) \) and increase on \( (-4, \infty) \).

### Summary:

- To obtain the graph of \( f \), shift the graph of \( y = \frac{1}{x} \) **4 units to the left**.
- **Domain:** \( (-\infty, -4) \cup (-4, \infty) \)
- **Range:** \( (-\infty, 0) \cup (0, \infty) \)
- **Intervals of Decrease:** \( (-\infty, -4) \)
- **Intervals of Increase:** \( (-4, \infty) \)

Would you like more details or have any questions about this solution? 

Here are five related questions to expand your understanding:
1. How would the graph change if we had \( f(x) = \frac{1}{x-4} \)?
2. What are the horizontal and vertical asymptotes of the function \( g(x) = \frac{1}{x-3} \)?
3. How do you find the domain of a rational function?
4. What is the effect of a vertical shift on the graph of \( y = \frac{1}{x} \)?
5. How do you determine the intervals where a function is increasing or decreasing?

**Tip:** When analyzing transformations of functions, always identify the basic function and apply the transformations step-by-step (shifts, stretches, reflections, etc.).

Explain how the graph of the function f(x) = 1/(x+4) can be obtained from the graph of y = 1/x. Then graph f and give the (a) domain and (b) range. Determine the largest open intervals of the domain over which the function is (c) increasing or (d) decreasing.

Learn how to transform the graph of the function f(x) = 1/(x+4) from y = 1/x, including determining the domain, range, and intervals where the function is increasing or decreasing. This solution is suitable for high school students in Grades 10-12 studying rational functions and graph transformations.

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