MPT Case Analysis 2 — Questions and Answers
Question 1: A piecewise function is defined as f(x) = 3x + 1 if x < 2, and f(x) = x² - 1 if x ≥ 2. What is f(2) + f(0)?
- 4 (Correct answer)
- 3
- 5
- 6
Correct answer: 4
f(2) = 2² - 1 = 3 (x ≥ 2 case), and f(0) = 3(0) + 1 = 1 (x < 2 case), so f(2) + f(0) = 4.
Question 2: Solve the absolute value inequality |2x - 4| > 6 and express as a compound inequality.
- x < -1 or x > 5 (Correct answer)
- x < 1 or x > 5
- -1 < x < 5
- x < -1 or x > 7
Correct answer: x < -1 or x > 5
Split into 2x - 4 > 6 (giving x > 5) or 2x - 4 < -6 (giving x < -1), so x < -1 or x > 5.
Question 3: A function g(x) = |x + 3| - |x - 1|. What is g(-5)?
- -4 (Correct answer)
- 4
- -6
- 6
Correct answer: -4
g(-5) = |-5 + 3| - |-5 - 1| = |-2| - |-6| = 2 - 6 = -4.
Question 4: For what values of x does the piecewise function h(x) = x² if x ≤ 0, and h(x) = 2x if x > 0 satisfy h(x) = 4?
- x = -2 only
- x = 2 only
- x = -2 and x = 2 (Correct answer)
- No solution
Correct answer: x = -2 and x = 2
For x ≤ 0: x² = 4 gives x = -2 (valid); for x > 0: 2x = 4 gives x = 2 (valid), so both solutions exist.
Question 5: A grading scale assigns A if s ≥ 90, B if 80 ≤ s < 90, and C if 70 ≤ s < 80. Which grade does a score of s = 80 receive?
- A
- B (Correct answer)
- C
- D
Correct answer: B
Since 80 ≤ 80 < 90, the score satisfies the B condition.
Question 6: What is the number of solutions to |x² - 4| = 0?
- 0
- 1
- 2 (Correct answer)
- 4
Correct answer: 2
The absolute value equals zero only when x² - 4 = 0, giving x = 2 or x = -2, so there are 2 solutions.
Question 7: The function f(x) = (x² - 9)/(x - 3) for x ≠ 3. What value should be assigned at x = 3 to make f continuous?
- 0
- 3
- 6 (Correct answer)
- 9
Correct answer: 6
Factor: (x² - 9)/(x - 3) = x + 3 for x ≠ 3; as x → 3 the limit is 3 + 3 = 6.
MPT Case Analysis 2
A piecewise function is defined as f(x) = 3x + 1 if x < 2, and f(x) = x² - 1 if x ≥ 2.
What is f(2) + f(0)?