https://file.aiquickdraw.com/mathbot/file/17271717437541sufvd0t.pdf

The document contains various mathematical problems related to inequalities, composite functions, function properties (like domain, range, even/odd), and transformations. Here is a summary of the main tasks found in the file:

1. **Solving Inequalities**: A series of inequalities need to be solved for their solution sets. For example, the inequalities listed include expressions like:
   \[
   2x + 1 \leq 5 \quad \text{and} \quad 3x - 4 > 2
   \]
   
2. **Composite Functions**:
   Given \( f(x) = x + 1 \) and \( g(x) = 2x - 3 \), you need to:
   - Find \( f \circ g \) (composite function),
   - Determine the domain of \( f \circ g \),
   - Compute \( (f \circ g)(3) \).

3. **Properties of Specific Functions**:
   - For \( f(x) = 1 - x^2 \), find:
     - The domain and range,
     - Determine whether the function is even, odd, or neither,
     - Investigate the monotonicity of the function.
     
   - For \( f(x) = \sqrt{1 - x^2} \), similar properties are required:
     - Domain and range,
     - Even or odd behavior,
     - Check if the function is bounded.

4. **Proof of Injectivity and Inverses**:
   You are asked to prove that a function \( f(x) = 2 - 3x \) is injective (one-to-one) and then find its inverse.

Let me know which specific problem you'd like to dive into, or if you'd like detailed solutions for all of them!

### Follow-up Questions:
1. How do we solve the inequality \( 2x + 1 \leq 5 \)?
2. What is the domain of the function \( g(x) = 2x - 3 \)?
3. How can we prove that a function is injective?
4. What are the properties of even and odd functions?
5. How do we find the inverse of \( f(x) = 2 - 3x \)?

**Tip**: Always check the domain restrictions when dealing with composite functions, as the range of the inner function must fit within the domain of the outer function.

أوجد مجموعة حلول المتباينات التالية: 2x + 1 ≤ 5 و 3x - 4 > 2. احسب التركيب (f ∘ g)(3) لدوال f(x) = x + 1 و g(x) = 2x - 3. حدد نطاق ومدى الدالة f(x) = 1 - x². أثبت أن الدالة f(x) = 2 - 3x تناظر آحادي وأوجد معكوسها.

Solve inequalities such as 2x + 1 ≤ 5 and explore composite functions with f(x) = x + 1 and g(x) = 2x - 3. Learn how to determine the domain and range of functions like f(x) = 1 - x² and prove injectivity for f(x) = 2 - 3x while finding its inverse. 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