https://r2.erweima.ai/mathbot/image/1725468890777xio0pzqj.jpg

 (a) define the domain (b) use the continuity theorem to test the continuity at each gaps in the domain (c) Use limits to determine the end end behavior and eventual asympotes (d) verify your results using your graphic calculator (B) Ex 59-62

To address the tasks presented, let's go through the problems step by step:

### (a) Define the Domain
The domain of a function is the set of all possible input values (x-values) that will produce a valid output from the function. I'll go through each function from the image to determine the domain.

1. **\( f(x) = -3x^2 + 7 \)**
   - Domain: \( \mathbb{R} \) (all real numbers)
   
2. **\( f(x) = \frac{x^2 - 2}{x} \)**
   - Domain: \( x \neq 0 \)
   
3. **\( f(x) = |x + 3| \)**
   - Domain: \( \mathbb{R} \) (all real numbers)
   
4. **\( f(x) = \frac{3x^2 - x - 2}{x - 1} \)**
   - Domain: \( x \neq 1 \)
   
5. **\( f(x) = \begin{cases} \frac{3x^2 - x - 2}{x - 1}, & x \neq 1 \\ 0, & x = 1 \end{cases} \)**
   - Domain: \( \mathbb{R} \) (all real numbers)

6. **\( f(x) = \begin{cases} \frac{5 - x}{2x - 3}, & x \leq 2 \\ x > 2 \end{cases} \)**
   - Domain: \( x \leq 2 \cup x > 2 \) (all real numbers except \( x = 1.5 \) which makes the denominator zero in the first piece)
   
7. **\( f(x) = \frac{1}{(x - 2)^2} \)**
   - Domain: \( x \neq 2 \)
   
8. **\( f(x) = \frac{3}{x + 1} \)**
   - Domain: \( x \neq  -1 \)
   
9. **\( f(x) = \frac{\sqrt{x + 1}}{x} \)**
   - Domain: \( x > 0 \)
   
10. **\( f(x) = \frac{x + 1}{2x + 2} \)**
    - Domain: \( x \neq -1 \)

### (b) Use the Continuity Theorem to Test the Continuity at Each Gap in the Domain

To apply the continuity theorem:
1. A function is continuous at a point if:
   - The function is defined at that point.
   - The limit as \( x \) approaches that point exists.
   - The function's value at that point equals the limit.

For each gap in the domain:
- **Discontinuities are expected at points where the denominator of the function equals zero** because the function is not defined there.

### (c) Use Limits to Determine the End Behavior and Eventual Asymptotes

For the limit analysis:
- **Horizontal Asymptotes** are found by evaluating the limit as \( x \) approaches infinity or negative infinity.
- **Vertical Asymptotes** are found by evaluating the limit as \( x \) approaches the points where the denominator is zero.

I'll now analyze one of the functions to provide an example.

**Example:** \( f(x) = \frac{3}{x + 1} \)

- **Vertical Asymptote:** Find \( x \) such that \( x + 1 = 0 \). So, \( x = -1 \).
  - The vertical asymptote is at \( x = -1 \).
- **Horizontal Asymptote:** As \( x \) approaches infinity or negative infinity, \( \lim_{x \to \pm \infty} \frac{3}{x + 1} = 0 \).
  - The horizontal asymptote is at \( y = 0 \).

### (d) Verify Your Results Using Your Graphic Calculator

For each function, use a graphing calculator to:
1. Plot the function.
2. Identify any discontinuities (gaps) or asymptotes.
3. Compare the graphical results with the analytical findings from parts (a), (b), and (c).

### Next Steps
I recommend doing this analysis for each function. If you'd like, I can continue solving for the remaining functions, or if you prefer, you can upload further functions or ask any specific questions.

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### Questions for Further Exploration:

1. How do you find the limit at infinity for a polynomial divided by another polynomial?
2. What happens to the function \( f(x) \) when the denominator approaches zero?
3. How do piecewise functions affect continuity and limits?
4. How do you determine the horizontal asymptote for rational functions?
5. What role does the absolute value play in determining the domain of \( f(x) = |x + 3| \)?

**Tip:** When analyzing the domain and continuity of a function, always look for values of \( x \) that make the denominator zero or involve square roots of negative numbers. These are usually where gaps or discontinuities occur.

(a) Define the domain (b) Use the continuity theorem to test the continuity at each gap in the domain (c) Use limits to determine the end behavior and eventual asymptotes (d) Verify your results using your graphic calculator.

Learn how to determine the domain of various functions, test their continuity, find limits to analyze end behavior, and identify asymptotes. Suitable for students in Grades 10-12.

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