https://file.aiquickdraw.com/mathbot/file/17266774131893jfhinuu.jpg

The task in the image asks to "Determine the domain of each of the following functions, given their representative curves."

From the two graphs, we can deduce the domain (ensemble de définition) based on where the functions are defined.

1. **Top Graph**:
   - The curve exhibits a vertical asymptote at \(x = 1\).
   - The function appears continuous for \(x < 1\) and \(x > 1\).
   - **Domain**: The function is defined for \(x \in \mathbb{R} \setminus \{1\}\), meaning all real numbers except \(x = 1\).

2. **Bottom Graph**:
   - This graph shows a vertical asymptote at \(x = -1\).
   - The function is continuous for \(x < -1\) and \(x > -1\).
   - **Domain**: The function is defined for \(x \in \mathbb{R} \setminus \{-1\}\), meaning all real numbers except \(x = -1\).

### In summary:
- **First function**: Domain \(x \in \mathbb{R} \setminus \{1\}\).
- **Second function**: Domain \(x \in \mathbb{R} \setminus \{-1\}\).

Would you like more detailed explanations, or do you have any further questions?

Here are some follow-up questions to deepen your understanding:
1. What is the general behavior of functions with vertical asymptotes?
2. How do you determine the domain of a function from its graph?
3. Can a function have more than one vertical asymptote? What would that imply for its domain?
4. What does a vertical asymptote tell you about the limit of a function as \(x\) approaches the asymptote?
5. How can discontinuities in a function's graph affect its domain?

**Tip**: When analyzing a graph to determine a function's domain, always look for points of discontinuity, such as vertical asymptotes, holes, or gaps in the curve.

Déterminer l'ensemble de définition de chacune des fonctions données par leurs courbes représentatives.

Learn how to determine the domain of functions based on their representative curves, focusing on identifying vertical asymptotes and continuous intervals. This problem helps students analyze function behavior visually, a key skill for Grades 9-12 math.

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