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.).