Let's work through each question in detail.

Question 1

Let:

• $A$: The set of the first 8 even numbers
• $B$: The set of the first 8 multiples of 3

1. Write $A$ and $B$ in Roster Method:

   • The first 8 even numbers are: $2, 4, 6, 8, 10, 12, 14, 16$

   • So, $A = \{ 2, 4, 6, 8, 10, 12, 14, 16 \}$

   • The first 8 multiples of 3 are: $3, 6, 9, 12, 15, 18, 21, 24$

   • So, $B = \{ 3, 6, 9, 12, 15, 18, 21, 24 \}$

2. Verify that $A \cap B = B \cap A$:

   • The intersection of $A$ and $B$ (common elements) is $\{ 6, 12 \}$
   • So, $A \cap B = \{ 6, 12 \}$ and $B \cap A = \{ 6, 12 \}$
   • Therefore, $A \cap B = B \cap A$.

Question 2

Let $z = 3 - 7i$. Find $\left| \frac{z}{\bar{z}} \right|$, where $\bar{z}$ is the complex conjugate of $z$.

1. The complex conjugate of $z = 3 - 7i$ is $\bar{z} = 3 + 7i$.
2. Calculate $\frac{z}{\bar{z}}$: $\frac{z}{\bar{z}} = \frac{3 - 7i}{3 + 7i}$ Multiply the numerator and denominator by the conjugate of the denominator: $\frac{(3 - 7i)(3 - 7i)}{(3 + 7i)(3 - 7i)} = \frac{9 - 42i + 49i^2}{9 - 49i^2} = \frac{9 - 42i - 49}{9 + 49} = \frac{-40 - 42i}{58} = -\frac{20}{29} - \frac{21}{29}i$
3. Calculate the magnitude: $\left| \frac{z}{\bar{z}} \right| = \sqrt{\left( -\frac{20}{29} \right)^2 + \left( -\frac{21}{29} \right)^2} = \sqrt{\frac{400}{841} + \frac{441}{841}} = \sqrt{\frac{841}{841}} = 1$ So, $\left| \frac{z}{\bar{z}} \right| = 1$.

Question 3

Let $f(x) = 5x + 1$ and $g(x) = x^2$. Find the composite functions $(f \circ g)(x)$ and $(g \circ f)(x)$.

1. Find $(f \circ g)(x)$:

   • $(f \circ g)(x) = f(g(x)) = f(x^2) = 5(x^2) + 1 = 5x^2 + 1$

2. Find $(g \circ f)(x)$:

   • $(g \circ f)(x) = g(f(x)) = g(5x + 1) = (5x + 1)^2 = 25x^2 + 10x + 1$

Question 4

Solve the quadratic equation $2x^2 - 4x + 1 = 0$.

1. Using the quadratic formula: $x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4 \cdot 2 \cdot 1}}{2 \cdot 2}$ Simplify inside the square root: $x = \frac{4 \pm \sqrt{16 - 8}}{4} = \frac{4 \pm \sqrt{8}}{4} = \frac{4 \pm 2\sqrt{2}}{4} = 1 \pm \frac{\sqrt{2}}{2}$ So, the solutions are: $x = 1 + \frac{\sqrt{2}}{2} \quad \text{and} \quad x = 1 - \frac{\sqrt{2}}{2}$

Would you like further details or explanations on any specific question?

Here are five related questions to explore further:

1. How do you find the union of two sets, like $A \cup B$, and how is it different from the intersection?
2. How can you generalize the concept of complex conjugates for any complex number?
3. What happens when you reverse the order in composite functions? Does it always yield a different result?
4. How can we solve a quadratic equation by completing the square?
5. What are some practical applications of complex numbers in real-world problems?

Tip: For composite functions, remember that $(f \circ g)(x)$ means applying $g$ first, then $f$. This order matters and often leads to different results from $(g \circ f)(x)$.