Step Function Intuition Math Example 1

Follow the full solution, then compare it with the other examples linked below.

Example 1

easy
Evaluate the floor function f(x)=⌊x⌋f(x) = \lfloor x \rfloor at x=3.7x = 3.7, x=−2.1x = -2.1, and x=5x = 5. Then describe the graph on [0,4][0, 4].

Solution

  1. 1
    ⌊3.7⌋=3\lfloor 3.7 \rfloor = 3 (greatest integer ≤3.7\leq 3.7).
  2. 2
    ⌊−2.1⌋=−3\lfloor -2.1 \rfloor = -3 (greatest integer ≤−2.1\leq -2.1 is −3-3, not −2-2).
  3. 3
    ⌊5⌋=5\lfloor 5 \rfloor = 5. Graph on [0,4][0,4]: horizontal steps at heights 0,1,2,30,1,2,3. Each step spans a half-open interval [n,n+1)[n, n+1) with a closed left endpoint and open right endpoint.

Answer

⌊3.7⌋=3\lfloor 3.7\rfloor=3; ⌊−2.1⌋=−3\lfloor -2.1\rfloor=-3; ⌊5⌋=5\lfloor 5\rfloor=5
The floor function always rounds down toward −∞-\infty. For negative numbers, this means ⌊−2.1⌋=−3\lfloor -2.1\rfloor=-3, not −2-2, because −3≤−2.1<−2-3 \leq -2.1 < -2.

About Step Function Intuition

A step function is piecewise constant — it takes a fixed value on each of several intervals, jumping abruptly at the interval boundaries.

Learn more about Step Function Intuition →

More Step Function Intuition Examples

Example 2 medium

A parking garage charges [formula]3[formula][formula] for each additional hour (or part). Write and

Example 3 easy

Evaluate: (a) [formula], (b) [formula], (c) [formula], (d) [formula].

Example 4 hard

Define [formula]. Find all [formula] in [formula] where [formula], and sketch [formula] on [formula]

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Related Concepts

Piecewise Function