1. គេមានអនុគមន៍ f មួយក្រាប ( C) តំណាងដោយ y= f(x) = 2(x+1)/( x+2) ។ 
 ក. រកដែនកំណត់នៃអនុគមន៍ f ។ 
 ខ. រកលីមីតចុងដែននៃអនុគមន៍ ។ 
 គ. ទាញរកសមីការ អាសុីមតូតឈរ និង អាសុីមតូតដេក ។ 
 ឃ. គណនា f'(x) និងសង់តារាង អថេរភាព f'(x) ។ 
 ង. សង់តារាងអថេរភាព ។ 
 ឃ. សង់ក្រាប graph 📉. 
 Please tell me 

Let's go through each part of the problem step by step.

### Given Function:
\[ f(x) = \frac{2(x+1)}{x+2} \]

### ក. Domain of the Function \( f(x) \)

To find the domain of \( f(x) \), we need to determine the values of \( x \) for which the function is defined. The function is undefined when the denominator is zero.

The denominator is \( x + 2 \). Set it to zero:
\[ x + 2 = 0 \]
\[ x = -2 \]

So, the domain of \( f(x) \) is all real numbers except \( x = -2 \).
**Domain**: \( \mathbb{R} \setminus \{-2\} \) or \( (-\infty, -2) \cup (-2, \infty) \)

### ខ. Limits at the Endpoints of the Domain (Limits at Infinity)

#### 1. Limit as \( x \) approaches \( -2 \) from the right:
\[ \lim_{{x \to -2^+}} \frac{2(x+1)}{x+2} \]
As \( x \) approaches \( -2 \) from the right, the denominator approaches \( 0^+ \), which implies the fraction tends towards \( +\infty \).

#### 2. Limit as \( x \) approaches \( -2 \) from the left:
\[ \lim_{{x \to -2^-}} \frac{2(x+1)}{x+2} \]
As \( x \) approaches \( -2 \) from the left, the denominator approaches \( 0^- \), which implies the fraction tends towards \( -\infty \).

#### 3. Limit as \( x \) approaches infinity:
\[ \lim_{{x \to \infty}} \frac{2(x+1)}{x+2} \]
Dividing both numerator and denominator by \( x \):
\[ = \lim_{{x \to \infty}} \frac{2(1 + \frac{1}{x})}{1 + \frac{2}{x}} = \frac{2(1 + 0)}{1 + 0} = 2 \]

#### 4. Limit as \( x \) approaches negative infinity:
\[ \lim_{{x \to -\infty}} \frac{2(x+1)}{x+2} \]
Similarly, as \( x \) approaches \( -\infty \), the limit also approaches \( 2 \).

### គ. Asymptotes (Vertical and Horizontal)

#### 1. Vertical Asymptote:
The vertical asymptote occurs where the function is undefined, which we found at \( x = -2 \).
**Vertical Asymptote**: \( x = -2 \)

#### 2. Horizontal Asymptote:
The horizontal asymptote is the limit of the function as \( x \) approaches \( \infty \) or \( -\infty \). Both limits are \( 2 \).
**Horizontal Asymptote**: \( y = 2 \)

### ឃ. Derivative \( f'(x) \) and Table of Signs

Let's calculate the derivative using the quotient rule:
\[ f(x) = \frac{2(x+1)}{x+2} \]

The quotient rule states:
\[ f'(x) = \frac{(v \cdot u' - u \cdot v')}{v^2} \]
Where:
- \( u = 2(x+1) \)
- \( v = x+2 \)
- \( u' = 2 \)
- \( v' = 1 \)

Applying this:
\[
f'(x) = \frac{(x+2) \cdot 2 - 2(x+1) \cdot 1}{(x+2)^2} = \frac{2(x+2) - 2(x+1)}{(x+2)^2}
\]
\[
f'(x) = \frac{2x + 4 - 2x - 2}{(x+2)^2} = \frac{2}{(x+2)^2}
\]

The derivative \( f'(x) = \frac{2}{(x+2)^2} \) is always positive, indicating that the function is increasing everywhere on its domain.

**Table of Signs for \( f'(x) \)**:
- \( f'(x) > 0 \) for all \( x \neq -2 \) (function is increasing everywhere).

### ង. Behavior of \( f(x) \)

Since \( f'(x) > 0 \), the function \( f(x) \) is strictly increasing over its entire domain \( (-\infty, -2) \cup (-2, \infty) \).

### ឃ. Graph of the Function \( f(x) \)

The graph of \( f(x) \) has:
- A vertical asymptote at \( x = -2 \)
- A horizontal asymptote at \( y = 2 \)
- The function is increasing for all \( x \neq -2 \)

Would you like me to generate the graph of this function for you?

### Would you like more details or have any questions?

Here are 8 related questions you might consider:
1. What is the significance of asymptotes in graphing functions?
2. How do you find the derivative of a function using the quotient rule?
3. What are the different methods for finding limits?
4. How do you determine whether a function is increasing or decreasing?
5. What is the relationship between the derivative and the slope of a function?
6. How can you graph a rational function step by step?
7. How do vertical and horizontal asymptotes influence the shape of a graph?
8. What are common challenges in finding the domain of a function?

**Tip:** When graphing a rational function, always check for asymptotes and behavior at infinity to understand the overall shape.

Explore the function f(x) = 2(x+1)/(x+2) with detailed solutions on domain, limits, derivatives, asymptotes, and graphing. Learn about the behavior of rational functions and how to apply limit theorems. Suitable for advanced high school students studying calculus.

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