Step Function Intuition Math Example 4

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

Example 4

hard
Define f(x)=⌊2x⌋f(x) = \lfloor 2x \rfloor. Find all xx in [0,2][0, 2] where f(x)=3f(x) = 3, and sketch ff on [0,2][0,2].

Solution

  1. 1
    Solve ⌊2x⌋=3\lfloor 2x \rfloor = 3: this requires 3≤2x<4⇒32≤x<23 \leq 2x < 4 \Rightarrow \frac{3}{2} \leq x < 2.
  2. 2
    So f(x)=3f(x)=3 on [32,2)[\frac{3}{2}, 2). Sketch: steps at heights 0,1,2,30,1,2,3 over [0,2][0,2], each step of width 12\frac{1}{2}: [0,12),[12,1),[1,32),[32,2)[0,\frac{1}{2}), [\frac{1}{2},1), [1,\frac{3}{2}), [\frac{3}{2},2).

Answer

f(x)=3f(x) = 3 for x∈[32,2)x \in \left[\dfrac{3}{2}, 2\right)
Scaling the argument of a floor function by 22 doubles the frequency of the steps (halves the step width). Each step now has length 12\frac{1}{2} instead of 11.

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 1 easy

Evaluate the floor function [formula] at [formula], [formula], and [formula]. Then describe the grap

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].

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Related Concepts

Piecewise Function