MPT Case Analysis 5

Question 1

A function f is defined as f(x) = 1/(x - 2) for x > 2, and f(x) = x - 5 for x ≤ 2. What is f(2) + f(4)?

Accepted Answer

-2.5

Answer

f(2) = 2 - 5 = -3 (x ≤ 2 piece); f(4) = 1/(4-2) = 0.5 (x > 2 piece); sum = -3 + 0.5 = -2.5.

Question 2

Case analysis on absolute value: solve |5 - 2x| ≤ 3.

Accepted Answer

1 ≤ x ≤ 4

Answer

-3 ≤ 5 - 2x ≤ 3 → -8 ≤ -2x ≤ -2 → dividing by -2 and flipping: 1 ≤ x ≤ 4.

Question 3

A piecewise function has f(x) = 4 for all x. Which statement correctly describes its case structure?

Accepted Answer

It can be written as piecewise with identical pieces on each region

Answer

Any function can be written piecewise; a constant function simply uses the same value in every case, though this adds no information.

Question 4

A savings plan adds $100/month for balances below $1,000 and $150/month for balances ≥ $1,000. Starting at $950, what is the balance after 2 months?

Accepted Answer

$1,200

Answer

Month 1: $950 < $1,000 → add $100 → $1,050; Month 2: $1,050 ≥ $1,000 → add $150 → $1,200.

Question 5

For f(x) = x + 1 if x < 1 and f(x) = x² if x ≥ 1, is f continuous at x = 1?

Accepted Answer

No, because the left limit is 2 but f(1) = 1

Answer

Left limit: lim(x→1⁻) x+1 = 2; actual value: f(1) = 1² = 1; since 2 ≠ 1, f is not continuous at x = 1.

(MPT) UCF Math Placement Test Practice Test