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From the photo, I can see a page with handwritten solutions related to functions, their domains, and solving equations for rational functions. Let me break down the contents:

### 3. Equation:
The equation \( g(a) = \frac{a(a - 1)}{a^2 - 4} \) has been simplified to solve for \( a \). After simplifying, the equation turns into \( a^2 - 4 = 0 \), and factoring gives:
\[
(a - 2)(a + 2) = 0
\]
So the solutions are \( a = 2 \) and \( a = -2 \). The domain of this function excludes \( a = 2 \) and \( a = -2 \), as they make the denominator zero. Thus, the **domain** is written as:
\[
\text{Domain: } a \in (-\infty, -2) \cup (-2, 2) \cup (2, \infty)
\]

### 4. Equation:
For the function \( y = \frac{4x^2 + 4x + 1}{x^2 - 10x - 24} \), you need to solve for when the denominator is zero by setting \( x^2 - 10x - 24 = 0 \). This quadratic equation factors into:
\[
(x - 12)(x + 2) = 0
\]
Thus, the solutions are \( x = 12 \) and \( x = -2 \). These values are excluded from the domain, so the domain is written as:
\[
\text{Domain: } x \in (-\infty, -2) \cup (-2, 12) \cup (12, \infty)
\]

### 5. Equation:
The function \( y = \frac{5x + 1}{x^2 - 10x - 24} \) also involves finding when the denominator equals zero, which is the same quadratic as in question 4:
\[
x^2 - 10x - 24 = 0
\]
The factored form is \( (x - 12)(x + 2) = 0 \), so the domain excludes \( x = 12 \) and \( x = -2 \). Therefore, the **domain** is:
\[
\text{Domain: } x \in (-\infty, -2) \cup (-2, 12) \cup (12, \infty)
\]

### 6. Equation:
For \( f(x) = \frac{2x}{x - 11} \), you need to find when the denominator equals zero. Set \( x - 11 = 0 \), which gives \( x = 11 \). Therefore, the function is undefined at \( x = 11 \), and the **domain** is:
\[
\text{Domain: } x \in (-\infty, 11) \cup (11, \infty)
\]

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### Related Questions:
1. How do you find the domain of a rational function in general?
2. What is the significance of factoring in solving quadratic equations?
3. How can you identify vertical asymptotes from a rational function?
4. What are the steps to solve a quadratic equation by factoring?
5. How does excluding values from the domain affect the behavior of the function?

### Tip:
When solving rational equations, always check the denominator for zero values, as these will define the restrictions on the domain!

Find the domains of the given rational functions and solve for their excluded values.

Learn how to find the domain of rational functions by solving quadratic equations in the denominator. This lesson covers algebraic factoring techniques and domain restrictions for rational functions, suitable for Grades 9-12 students.

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