https://file.aiquickdraw.com/mathbot/file/17278034745411mh5b5w7.jpg

Here is a transcription and breakdown of the problems from the image you uploaded:

---

### 1. Solve the following inequalities and express the solution set in three notations (interval notation, set notation, and on a number line):
   a. \( x^2 - 2x + 2 \leq 6x - x^2 - 3 \)
   b. \( |2x + 3| > |x - 3| \)

### 2. If \( f(x) = x^2 \), sketch the graph of the function \( g(x) = 3x^2 - 12x + 16 \) by transforming the graph of \( f(x) \).

### 3. Given \( f(x) = \frac{2x}{x - 3} \) and \( g(x) = \sqrt{2x} - 5 \), determine \( (f \circ g)(x) \) and the domain \( D_{f \circ g} \).

### 4. Prove that \( \lim_{x \to 2} 3x + 2 = 8 \).

### 5. Compute \( \lim_{t \to 0} \frac{\sin(2t) + 5t}{t \cdot \sec(t)} \).

### 6. Determine the horizontal and vertical asymptotes of the function:
   a. \( f(x) = \frac{6x^2 - 30x + 36}{2x^2 - 2x - 12} \)

### 7. Determine where the following function is not continuous:
   \[
   f(x) = 
   \begin{cases} 
   2x^2 - 1, & x < -2 \\
   2x - 2, & -2 \leq x < 2 \\
   3x - x^2, & x \geq 2
   \end{cases}
   \]

---

Would you like me to solve any of these problems in particular? Let me know if you'd like a detailed solution or have any specific questions!

Here are 5 related questions to expand on these problems:
1. What are the steps involved in solving quadratic inequalities?
2. How do transformations like translation and scaling affect the graph of a function?
3. How do you compute the composition of two functions, and what affects their domain?
4. What are the conditions for proving a limit using the epsilon-delta definition?
5. How do you determine the vertical and horizontal asymptotes of rational functions?

**Tip**: When working with inequalities involving absolute values, it's helpful to break them into different cases to handle the positive and negative parts separately.

Solve inequalities, graph transformations, function compositions, limits, and asymptotes from the provided mathematical problems.

This page covers solving inequalities, understanding graph transformations, computing function compositions, finding limits, and determining asymptotes and continuity for functions. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation