https://r2.erweima.ai/mathbot/image/17243744201300696x0fp.jpg

The image contains a problem regarding the conditions of existence for a function of a real variable, presented in Spanish. The task is to indicate the conditions of existence for the function \( f(x) \) given in different cases.

### Detailed Translation and Interpretation:

1. **a)** La función \( f(x) \) está definida en el punto \( x_0 = 2 \cdot \text{sen}(3x) \) si y sólo si \( 3x \) está definido y existe \( \forall x \in [0,1] \) que se verifique que \( \text{log} (10x) > 0 \).

   - *Translation:* The function \( f(x) \) is defined at the point \( x_0 = 2 \cdot \sin(3x) \) if and only if \( 3x \) is defined and exists \( \forall x \in [0,1] \) such that \( \log(10x) > 0 \).

2. **b)** La función \( f(x) \) se continúa en el punto \( x_0 = \sqrt[3]{x^2} \) si y sólo si \( \log(5x) \) está definido y existe \( \forall x \in [-1,1] \).

   - *Translation:* The function \( f(x) \) is continuous at the point \( x_0 = \sqrt[3]{x^2} \) if and only if \( \log(5x) \) is defined and exists \( \forall x \in [-1,1] \).

3. **c)** La función \( f(x) \) es continua en el punto \( x_0 = f(5x^2) \) si y sólo si \( f(5x^2) \) está definido y existe \( \forall x \in [-1,1] \).

   - *Translation:* The function \( f(x) \) is continuous at the point \( x_0 = f(5x^2) \) if and only if \( f(5x^2) \) is defined and exists \( \forall x \in [-1,1] \).

4. **d)** La función \( f(x) \) se continúa en el punto \( x_0 = f(\text{ln}(x^2 - 1)) \) si y sólo si \( \log(\sqrt{x^2 - 1}) \) está definida y existe \( \forall x > 0 \).

   - *Translation:* The function \( f(x) \) is continuous at the point \( x_0 = f(\ln(x^2 - 1)) \) if and only if \( \log(\sqrt{x^2 - 1}) \) is defined and exists \( \forall x > 0 \).

### Answering the Questions:
To determine the conditions for the existence and continuity of the function at the given points, we analyze the definitions provided in the problem:

1. **For part (a):**
   - The function involves \( \log(10x) > 0 \).
   - \( \log(10x) > 0 \) implies \( 10x > 1 \) or \( x > \frac{1}{10} \).
   - Therefore, \( x \) must be in the interval \( \left(\frac{1}{10}, 1\right] \) for the function to be defined.

2. **For part (b):**
   - \( \log(5x) \) must be defined.
   - \( \log(5x) \) exists if \( 5x > 0 \) or \( x > 0 \).
   - The interval provided is \( x \in [-1, 1] \), so for continuity, \( x \) must be within \( (0, 1] \).

3. **For part (c):**
   - The expression involves \( f(5x^2) \) and is defined in \( [-1, 1] \).
   - Since \( 5x^2 \) is always non-negative, \( f(5x^2) \) should be defined for \( x \) in this range.

4. **For part (d):**
   - The condition \( \log(\sqrt{x^2 - 1}) \) requires that \( x^2 - 1 > 0 \), so \( x > 1 \).
   - Therefore, the function is continuous if \( x > 1 \).

### Do you want details or have any specific questions?

### Relative Questions:
1. What are the implications if \( \log(10x) \) was non-positive?
2. How would the continuity condition change if the intervals for \( x \) were different?
3. Can you analyze how the continuity condition changes for different logarithmic bases?
4. What happens if \( x = 1 \) for part (d)? Is the function defined or continuous?
5. How does the square root function affect the continuity in part (b)?

### Tip:
When analyzing functions involving logarithms, ensure the argument inside the logarithm is strictly positive to determine the domain of definition.

Explore detailed conditions for the existence and continuity of functions in various scenarios. Understand how logarithmic functions and specific function definitions affect continuity. Suitable for advanced high school students and beyond.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation