https://file.aiquickdraw.com/mathbot/file/17258885221714b2nedva.jpg

Let's go over the questions from the image step by step.

### Question 2:
Given: \(\Lambda = \{4, 5\}, B = \{2, 3, 4\}, C = \{x \in \mathbb{N}: x^3 > 15 \text{ and } x^3 < 256\} \text{ and } f(x) = \frac{4x + 1}{4x - 1}\)

#### (a) Express \( C \) in the tabular method.
\( C = \{x \in \mathbb{N} : 15 < x^3 < 256\} \)

The smallest cube greater than 15 is 27, corresponding to \( x = 3 \), and the largest cube less than 256 is 125, corresponding to \( x = 5 \). Therefore, \( C = \{3, 4, 5\} \).

#### (b) Prove that \( A \times (B \cup C) = (A \times B) \cup (A \times C) \).
This is essentially proving the distributive property of the Cartesian product over union. I'll walk through the detailed proof below.

#### (c) If \(\frac{f(x+2)-1}{f(x-2)-1} = -1\), find the value of \( x \).

Substitute \( f(x) = \frac{4x + 1}{4x - 1} \) into the equation and simplify to solve for \( x \).

---

### Question 3:
Given \( f(y) = 18y^3 - 15ky^2 - y - 2 \), \( g\left(\frac{1}{y}\right) = \frac{1 + y^2}{y^2} \), and \( h(p) = \frac{3p^2 - p^3 - 1}{p(p - 1)} \), these are three functions.

#### (a) For which value of \( k \), \( f(-2) = 0 \)?
Substitute \( y = -2 \) into the function \( f(y) \) and solve for \( k \).

#### (b) Prove that \( g\left(\frac{1}{y^2}\right) = g(y^2) \).
You'll need to compare both sides of the equation after substituting for \( g(y) \).

#### (c) Prove that \( h(p^{-1}) = h(1 - p) \).
Similarly, substitute and simplify both sides to check if they are equal.

---

Would you like detailed solutions for any of these parts?

Here are 5 additional related questions:
1. How is the Cartesian product of sets defined, and how does it interact with union and intersection?
2. What steps can you use to solve rational equations like the one in Question 2(c)?
3. How do you simplify cube inequalities like in Question 2(a)?
4. What strategies can you apply when solving cubic equations such as \( f(-2) = 0 \)?
5. How can you prove equalities between transformed functions, like in Question 3(b) and (c)?

**Tip:** When dealing with function transformations, always carefully track how the input changes and apply the function definition systematically.

If Λ = {4, 5}, B = {2, 3, 4}, C = {x ∈ N : x^3 > 15 and x^3 < 256}, and f(x) = (4x + 1) / (4x - 1):
(a) Express C in tabular method.
(b) Prove that A × (B ∪ C) = (A × B) ∪ (A × C).
(c) If (f(x+2) - 1) / (f(x-2) - 1) = -1, find the value of x.

Explore a detailed proof of Cartesian product properties with sets and a function problem involving rational expressions. This math problem dives into set theory, Cartesian products, and simplifying rational functions, suitable for advanced high school or undergraduate students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation