Let's begin by understanding the transformation of the graph of the function $f(x) = \frac{1}{x+5}$ from the graph of $y = \frac{1}{x}$.

Step 1: Transformation of the Graph

The function $f(x) = \frac{1}{x+5}$ is a horizontal shift of the basic rational function $y = \frac{1}{x}$. Here's how the transformation works:

• The basic graph of $y = \frac{1}{x}$ has a vertical asymptote at $x = 0$ and a horizontal asymptote at $y = 0$.
• In the equation $f(x) = \frac{1}{x+5}$, the term $+5$ inside the denominator indicates a horizontal shift of the graph. Specifically, the graph of $y = \frac{1}{x}$ will be shifted 5 units to the left.

So, the transformation that occurs is a horizontal translation to the left by 5 units.

Step 2: Graph of $f(x) = \frac{1}{x+5}$

To graph $f(x)$, follow these steps:

• Start by graphing the basic function $y = \frac{1}{x}$.
• Move the entire graph 5 units to the left.
• The vertical asymptote of $y = \frac{1}{x}$ at $x = 0$ becomes $x = -5$ for $f(x)$.
• The horizontal asymptote remains $y = 0$.

The graph of $f(x)$ will have two branches:

• One in the first quadrant (as $x \to -5^+$), and the other in the third quadrant (as $x \to -5^-$).

Step 3: Domain and Range of $f(x)$

Domain: The domain of a rational function is all real numbers except where the denominator equals zero. For $f(x) = \frac{1}{x+5}$, the denominator equals zero when $x = -5$, so:

Domain: $(-\infty, -5) \cup (-5, \infty)$

Range: The range of $y = \frac{1}{x}$ is $(-\infty, 0) \cup (0, \infty)$. Since the transformation does not affect the range, the range of $f(x) = \frac{1}{x+5}$ is the same.

Range: $(-\infty, 0) \cup (0, \infty)$

Step 4: Intervals of Increase and Decrease

For the function $f(x) = \frac{1}{x+5}$, it behaves similarly to $y = \frac{1}{x}$, but shifted. The general behavior of the rational function $y = \frac{1}{x}$ is:

• Decreasing for $x > 0$.
• Increasing for $x < 0$.

Since the graph of $f(x) = \frac{1}{x+5}$ is a horizontal shift of $y = \frac{1}{x}$, we will have:

• Increasing for $x < -5$ (to the left of the vertical asymptote).
• Decreasing for $x > -5$ (to the right of the vertical asymptote).

Summary:

• Domain: $(-\infty, -5) \cup (-5, \infty)$
• Range: $(-\infty, 0) \cup (0, \infty)$
• Intervals of Increase: $(-\infty, -5)$
• Intervals of Decrease: $(-5, \infty)$

Graphing the Function

The graph of $f(x) = \frac{1}{x+5}$ can be sketched by shifting the graph of $y = \frac{1}{x}$ 5 units to the left.

Would you like me to generate the graph of $f(x)$ for you? Let me know if you have any questions or need further details!