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The task is to determine the largest domain (ensemble de définition) of the given functions. To do this, we will analyze where each function is defined by identifying the points where they are undefined.

### Given Functions:

1. \( f(x) = \frac{4 - x}{9x + 36} \)
2. \( g(x) = \frac{3}{x^2 + 7} \)
3. \( h(x) = \frac{3x}{4x^2 - 100} \)
4. \( i(x) = \frac{2x - 1}{(x - 8)(6x + 30)} \)

#### 1. Function \( f(x) \):
\[ f(x) = \frac{4 - x}{9x + 36} \]
To determine where \( f(x) \) is defined, we need to ensure that the denominator is not zero:
\[
9x + 36 \neq 0
\]
\[
9x \neq -36
\]
\[
x \neq -4
\]
Thus, \( f(x) \) is defined for all \( x \in \mathbb{R} \) except \( x = -4 \).

#### 2. Function \( g(x) \):
\[ g(x) = \frac{3}{x^2 + 7} \]
The denominator \( x^2 + 7 \) is always positive since \( x^2 \geq 0 \) for all \( x \). Therefore, \( g(x) \) is defined for all real numbers \( x \in \mathbb{R} \).

#### 3. Function \( h(x) \):
\[ h(x) = \frac{3x}{4x^2 - 100} \]
We need to find when the denominator is not zero:
\[
4x^2 - 100 = 0
\]
\[
4(x^2 - 25) = 0
\]
\[
x^2 - 25 = 0
\]
\[
(x - 5)(x + 5) = 0
\]
\[
x = 5 \quad \text{or} \quad x = -5
\]
Thus, \( h(x) \) is defined for all \( x \in \mathbb{R} \) except \( x = 5 \) and \( x = -5 \).

#### 4. Function \( i(x) \):
\[ i(x) = \frac{2x - 1}{(x - 8)(6x + 30)} \]
We need to find when the denominator is not zero:
\[
(x - 8)(6x + 30) = 0
\]
Solving each factor:
\[
x - 8 = 0 \quad \Rightarrow \quad x = 8
\]
\[
6x + 30 = 0 \quad \Rightarrow \quad 6x = -30 \quad \Rightarrow \quad x = -5
\]
Thus, \( i(x) \) is defined for all \( x \in \mathbb{R} \) except \( x = 8 \) and \( x = -5 \).

### Summary of Domains:
- \( f(x) \) is defined for \( x \in \mathbb{R} \setminus \{-4\} \)
- \( g(x) \) is defined for \( x \in \mathbb{R} \)
- \( h(x) \) is defined for \( x \in \mathbb{R} \setminus \{-5, 5\} \)
- \( i(x) \) is defined for \( x \in \mathbb{R} \setminus \{-5, 8\} \)

Would you like further details on any of these steps? Here are 5 related questions:

1. How do we determine the domain of more complex functions?
2. What happens when a function has multiple undefined points?
3. How would the domain change if a function involved square roots?
4. Can you explain the difference between the domain of a rational function and a polynomial?
5. How would asymptotes affect the domain of a function?

**Tip:** When solving for the domain of rational functions, always look for where the denominator equals zero, as these are the values excluded from the domain.

Déterminer le plus grand ensemble de définition des fonctions proposées: f(x) = (4 - x) / (9x + 36), g(x) = 3 / (x^2 + 7), h(x) = (3x) / (4x^2 - 100), i(x) = (2x - 1) / ((x - 8)(6x + 30))

Explore how to determine the largest domain of rational functions, analyzing cases where the denominator is zero. This includes functions like f(x) = (4 - x) / (9x + 36) and others. Learn step-by-step approaches and key concepts suitable for Grades 10-12.

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