Let's solve each problem in the image step-by-step.

Problem Solutions

48. Determine the interquartile range for the set {30, 48, 70, 77, 58, 69}.

1. Arrange the numbers in ascending order: 30, 48, 58, 69, 70, 77.
2. Find the quartiles:
   • Q1 (lower quartile) is the median of the first half: $Q1 = \frac{48 + 58}{2} = 53$.
   • Q3 (upper quartile) is the median of the second half: $Q3 = \frac{69 + 70}{2} = 69.5$.
3. Interquartile Range (IQR) = $Q3 - Q1 = 69.5 - 53 = 16.5$.

Answer: Closest option not given exactly, but expected answer should be near 17.

49. Transformation mapping points $(2, 2)$ to $(3, 3)$ and $(7, 4)$ to $(7, 6)$

• Observing the points, this transformation is consistent with Translation, where each point is shifted by the same vector.

Answer: B) Translation

50. Graph representing the equation $y = -2x + 1$

• The slope is -2, meaning it’s a negative slope (line slopes downward), and the y-intercept is $+1$.
• The correct graph is C, as it has a negative slope and a y-intercept of $+1$.

Answer: C

51. Which relation does NOT represent a function?

• A function must have unique x-values for each y-value.
• Relation C has repeated x-values (4) with different y-values (4 and 5), violating the rule of functions.

Answer: C

52. Given $f(x) = 2x + 5$ and $g(x) = x$, find $f(g(4))$.

1. First, find $g(4)$: $g(4) = 4$.
2. Now, substitute $g(4)$ into $f$: $f(4) = 2(4) + 5 = 8 + 5 = 13$.

Answer: D) 13

53. Find the inverse of $f(x) = \frac{x - 1}{3}$.

1. Set $y = \frac{x - 1}{3}$ and solve for $x$:
   • Multiply both sides by 3: $3y = x - 1$.
   • Add 1 to both sides: $x = 3y + 1$.
2. Replace $y$ with $f^{-1}(x)$: $f^{-1}(x) = 3x + 1$.

Answer: C) $f^{-1}(x) = 3x + 1$

54. Simplify the inequality $4(x - 6) \geq 6(x - 1)$.

1. Expand both sides: $4x - 24 \geq 6x - 6$.
2. Move terms with $x$ to one side: $-24 + 6 \geq 2x \Rightarrow -18 \geq 2x$.
3. Divide by 2: $-9 \geq x$ or $x \leq -9$.

Answer: A) $x \leq -9$

55. In the graph, given point $B(12, 5)$, find the likely coordinates of $A$.

• Observing the position, (8, 10) is in line with a plausible shift.

Answer: B) (8, 10)

56. The y-intercept of $y = 3x + 1$

• The y-intercept is the constant term when $x = 0$, which is $+1$.

Answer: B) 1

57. Line graph for $-2 \leq x \leq 3$.

• The correct representation is a closed circle at -2 and 3, representing inclusive boundaries.

Answer: A

Would you like further details on any problem?

Additional Questions:

1. How is the interquartile range calculated for other datasets?
2. How can we identify transformations between two sets of points?
3. What are other forms of functions and relations?
4. How do inverse functions relate to the original function?
5. What are strategies for solving linear inequalities?

Tip: When verifying if a relation is a function, ensure each x-value maps to only one y-value.