AMC 10 · 2019 · #9

Grade 8 arithmetic

Archetype Small-Domain Constrained Search └ Sub-archetype Logic-Deduction / Elimination

floor-functionabsolute-valuefunction-evaluationcasework caseworkpattern-recognition ↑ Prerequisites: floor-functionabsolute-value

📏 Medium solution 💡 3 insights

Problem

Define f(x) = ⌊|x|⌋ - |⌊ x ⌋| for every real x. Here ⌊ r ⌋ is the greatest integer ≤ r. What is the set of values f can take (its range)?

Pick an answer.

(A)

$\{-1, 0\}$

(B)

The set of nonpositive integers

(C)

$\{-1, 0, 1\}$

(D)

$\{0\}$

(E)

The set of nonnegative integers

AMC 10 2019 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Solve an Easier Related Problem

The formula has |·| and ⌊·⌋ that interact differently depending on whether x is positive, negative, integer, or non-integer (Tool #9: try one easy value from each kind of x). Tool #5 spots that within each kind, the answer is the same (no surprises from changing the size of x). Tool #6 plugs concrete numbers — 2.5, -2.5, -3, 0.5 — and Tool #3 then narrows the five answer choices to the one matching set of observed outputs.

1STEP 1

Case 1, x ≥ 0: the bars do nothing, so both terms match and f(x) = 0 (x = 2.5 → 2 - 2 = 0).

x ≥ 0 → f(x) = 0

2STEP 2

Case 2, x a negative integer (x = -3): |x| and ⌊ x ⌋ are already integers, so f(x) = 3 - 3 = 0.

x = -3 → ⌊ 3 ⌋ - | - 3| = 3 - 3 = 0

3STEP 3

Case 3, x a negative non-integer (x = -2.5): ⌊ 2.5 ⌋ = 2 but ⌊ -2.5 ⌋ = -3, so f = 2 - 3 = -1.

f(-2.5) = 2 - 3 = -1

4STEP 4

Pattern holds: x = -0.5 → 0 - 1 = -1, x = -10.2 → 10 - 11 = -1. Every negative non-integer gives -1.

x neg., non-integer → f(x) = -1

5STEP 5

All cases give only 0 (Cases 1-2) and -1 (Cases 3-4), so the range is {-1, 0}; (B)(C)(E) add unseen values, (D) misses -1.

Range(f) = {-1, 0} → (A)

Answer

{-1, 0}

The difference ⌊|x|⌋ - |⌊ x ⌋| measures how much the floor function disagrees with itself depending on order with absolute value. For x ≥ 0 these commute; for x a negative integer they still match; only for negative non-integers does floor go one extra integer down, producing a gap of exactly 1. So f takes only 0 or -1, never anything else — {-1, 0} is the full range, matching (A).

💡Key takeaway

This AMC 10 problem only needs Grade 8 function thinking you already know: split into positive vs. negative vs. integer cases. For x ≥ 0 and negative integers, f = 0; for negative non-integers (like -2.5), the floor drops one extra integer making f = -1. So the range is {-1, 0}.

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