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The image contains three math problems with subparts. Here is a breakdown of each question and the approach to solve them:

### **1. Determinar el dominio de las siguientes funciones:**
   1. \( f(x) = \frac{2x}{x + 5} \)
   2. \( f(x) = \sqrt{2x + 1} \)

### **2. Determine la continuidad de las siguientes funciones. Valore la función en \( x = 0 \):**
   1. \( f(x) = \cos(x + \frac{\pi}{2}) \)

### **3. Identifique la función que representa cada curva. Valore 15 puntos.**
   - The question shows three graphs labeled as (a), (b), and (c). The task is to identify the corresponding function for each graph.

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### **Solution Approach:**

#### **1. Determinar el dominio:**
   1. **\( f(x) = \frac{2x}{x + 5} \)**
      - The domain of a function is the set of all possible values of \( x \) for which the function is defined.
      - This function will be undefined if the denominator is zero, i.e., \( x + 5 = 0 \).
      - Solve for \( x \): \( x = -5 \).
      - So, the domain is \( \mathbb{R} \) except \( x = -5 \), or \( (-\infty, -5) \cup (-5, \infty) \).
   
   2. **\( f(x) = \sqrt{2x + 1} \)**
      - The domain is determined by the condition under which the expression inside the square root is non-negative, i.e., \( 2x + 1 \geq 0 \).
      - Solve for \( x \): \( x \geq -\frac{1}{2} \).
      - So, the domain is \( \left[-\frac{1}{2}, \infty\right) \).

#### **2. Determine la continuidad y valore en \( x = 0 \):**
   - **\( f(x) = \cos(x + \frac{\pi}{2}) \)**
     - This function is continuous everywhere because the cosine function is continuous over the entire real line.
     - To evaluate at \( x = 0 \), substitute \( x = 0 \) into the function:
     - \( f(0) = \cos\left(0 + \frac{\pi}{2}\right) = \cos\left(\frac{\pi}{2}\right) = 0 \).

#### **3. Identifique la función que representa cada curva:**
   - This part requires visual identification of the functions based on the provided graphs:
     - **(a)** appears to represent a quadratic function (like \( f(x) = x^2 \)).
     - **(b)** appears to represent a sine or cosine function, likely \( \sin(x) \) or \( \cos(x) \).
     - **(c)** resembles a cubic function (like \( f(x) = x^3 \)).

Would you like a more detailed explanation or have any questions about the solutions? 

### Related Questions:
1. What is the domain of \( f(x) = \sqrt{x^2 - 4x + 3} \)?
2. How do you determine the range of a function?
3. What are the conditions for a function to be continuous?
4. How do you identify the function based on its graph?
5. What is the derivative of \( f(x) = \cos(x + \frac{\pi}{2}) \)?
6. What is the significance of the square root in determining the domain?
7. How would the domain change if the function were \( f(x) = \frac{2x}{(x + 5)(x - 3)} \)?
8. How do you evaluate limits for functions involving square roots?

### Tip:
When determining the domain of a function, always look out for values that might cause the function to be undefined, such as division by zero or taking the square root of a negative number.

Aprende a resolver problemas matemáticos sobre determinación de dominio, continuidad de funciones y análisis gráfico. Incluye ejercicios detallados y soluciones paso a paso para identificar funciones a partir de gráficos. Adecuado para estudiantes de Grados 10-12.

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