Question 1: If (x/3 + 1, y - 2/3) = (5/3, 1/3), find the values of x and y.

Solution:

Since the ordered pairs are equal, their corresponding components must be equal.

1) Equating first components:

x/3 + 1 = 5/3

x/3 = 5/3 - 1 = 2/3

x = 2

2) Equating second components:

y - 2/3 = 1/3

y = 1/3 + 2/3 = 3/3

y = 1

Answer: x = 2 and y = 1

Question 2: If the set A has 3 elements and the set B = {3, 4, 5}, then find the number of elements in (A × B).

Solution:

Given: n(A) = 3

Set B = {3, 4, 5}, so n(B) = 3

We know that n(A × B) = n(A) × n(B)

n(A × B) = 3 × 3 = 9

Answer: The number of elements in (A × B) is 9.

Question 3: If G = {7, 8} and H = {5, 4, 2}, find G × H and H × G.

Solution:

1) G × H:

G × H = {(7, 5), (7, 4), (7, 2), (8, 5), (8, 4), (8, 2)}

2) H × G:

H × G = {(5, 7), (5, 8), (4, 7), (4, 8), (2, 7), (2, 8)}

Question 4: Find the domain and range of the real function f(x) = -|x|.

Solution:

Given function: f(x) = -|x|

1) Domain: |x| is defined for all real numbers x, so Domain = R (Set of Real Numbers).

2) Range: For any real number x, |x| ≥ 0. Multiplying by -1 gives -|x| ≤ 0.

Hence, f(x) yields only non-positive real values.

Range = (-∞, 0]

Question 5: Find the domain and range of the real function f(x) = √(9 - x²).

Solution:

1) Domain: For f(x) to be a real function, the quantity inside square root must be non-negative:

9 - x² ≥ 0 ⇒ x² ≤ 9 ⇒ -3 ≤ x ≤ 3

Domain = [-3, 3]

2) Range: For x ∈ [-3, 3], minimum value of y is √(9 - 3²) = 0 and maximum value is √(9 - 0²) = 3.

Range = [0, 3]

Question 6: A function f is defined by f(x) = 2x - 5. Write down the values of f(0), f(7), and f(-3).

Solution:

1) f(0) = 2(0) - 5 = 0 - 5 = -5

2) f(7) = 2(7) - 5 = 14 - 5 = 9

3) f(-3) = 2(-3) - 5 = -6 - 5 = -11

Question 7: Let f = {(1,1), (2,3), (0,-1), (-1,-3)} be a function from Z to Z defined by f(x) = ax + b. Determine a and b.

Solution:

Since (0, -1) ∈ f, we have f(0) = -1.

a(0) + b = -1 ⇒ b = -1

Since (1, 1) ∈ f, we have f(1) = 1.

a(1) + b = 1 ⇒ a + (-1) = 1 ⇒ a = 2

Answer: a = 2, b = -1