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The document you uploaded contains a "Taller de Cálculo Diferencial" for a technology program. Here’s a breakdown of the tasks outlined:

1. **Determine if the given diagrams and graphs represent a function.**
   - You are asked to analyze specific diagrams (A, B, M, N) and check if they fulfill the criteria for being functions.

2. **Determine whether given functions are even, odd, or neither:**
   - The functions provided are:
     - \( f(x) = \frac{x^3 - 2x}{x^3} \)
     - \( f(x) = x^3 - 2x \)
     - \( f(x) = (x + 1)^2 + x^3 \)
     - \( f(x) = \frac{x^2 - 1}{x^2 + 1} \)
     - \( f(x) = -x^2 + 2x - 4 \)

   The criteria for even and odd functions are:
   - Even: \( f(-x) = f(x) \)
   - Odd: \( f(-x) = -f(x) \)

3. **Find the domain of the following functions and graph them:**
   - Functions to work with:
     - \( f(x) = \frac{2x - 1}{x - 1} \)
     - \( f(x) = \sqrt{2x + 4} \)
     - \( g(x) = \frac{x}{3x - 1} \)
     - \( f(x) = \frac{x - 3}{2x^2 + 10x} \)
     - \( f(x) = \frac{1}{x^2 - 7x + 10} \)
     - \( f(x) = x^2 - 2x - 3 \)

4. **Evaluate the following functions:**
   - Evaluate \( f(x + h) - f(x) \) for \( f(x) = \frac{2x - 1}{2x + 1} \).
   - Find \( f(5) \) for \( f(x) = \sqrt{x^2 - 16} \).
   - Determine \( \frac{f(x + b) - f(x)}{b} \) for \( f(x) = \frac{1}{x + 1} \).
   - Evaluate \( f(-1) \) for \( f(x) = -3x^2 + \frac{2}{x^2} - \frac{3}{x} \).

Would you like me to solve specific problems from this list or provide further explanations? 

Here are five related questions that expand on the document:
1. How can you determine if a diagram or graph represents a function?
2. What is the method to find the domain of a rational function?
3. How do you prove if a function is even or odd algebraically?
4. What techniques are used to simplify the difference quotient \( \frac{f(x+h) - f(x)}{h} \)?
5. How can graphing help in verifying the domain of a function?

**Tip:** When determining the domain of a function, look for restrictions such as division by zero or square roots of negative numbers.

The document contains tasks involving analyzing diagrams to check if they represent functions, determining if functions are even or odd, finding the domain of functions and graphing them, and evaluating specific functions.

Explore detailed steps on analyzing whether diagrams represent functions, determining even or odd functions, finding the domain, and graphing rational functions. This problem includes key algebraic concepts and is suitable for high school students in Grades 10-12.

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