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: $\text{Domain: } x \in (-\infty, 11) \cup (11, \infty)$

Let me know if you would like further details on any of these steps!

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!