Question

How are step functions related to piecewise-defined functions?

How are the ceiling function and the floor function similar? How are they different?

Evaluate the ceiling function for the given value.
5. f(x) = [x]; x = 5.13
6. f(x) = ceiling(x); x = 11.71

Evaluate the floor function for the given value.
7. f(x = [x]; x = 9.37
8. f(x) = floor(x); x = 5.49
9. Graph the function f.
x f(x)
0 < x <= 1 4
1 < x <= 2 5
2 < x <= 3 6
3 < x <= 4 7
4 < x <= 5 8
5 < x <= 6 9

Answer 1

1) Relation between step functions and piecewise-defined functions
- A step function is a special case of a piecewise-defined function where each piece is a constant value on some interval.
- More generally, a piecewise-defined function can have different (nonconstant) formulas on different pieces; a step function just uses constant values on each piece.

2) How ceiling and floor are similar and different
- Similarities:
- Both map real numbers to integers.
- Both are monotone nondecreasing and are piecewise-constant (step) functions with jumps at integers.
- For any real x, floor(x) ≤ x ≤ ceiling(x), and at integers they are equal: floor(n) = ceiling(n) = n.
- Differences:
- floor(x) = greatest integer ≤ x. Example: floor(2.7) = 2.
- ceiling(x) = least integer ≥ x. Example: ceiling(2.7) = 3.
- Continuity: floor is right-continuous (value equals the right-hand limit at integers); ceiling is left-continuous (value equals the left-hand limit at integers).
- Relationship: floor(x) = −ceiling(−x).

Evaluations
(Note: the notation [x] is sometimes used for floor(x). Because your heading said “ceiling” for items 5–6, I treat them as ceiling evaluations below. If you instead meant [x] as floor, see the parenthetical alternatives.)

5. f(x) = [x]; x = 5.13
- If [x] means ceiling(x): ceiling(5.13) = 6.
- If [x] means floor(x): floor(5.13) = 5.

6. f(x) = ceiling(x); x = 11.71
- ceiling(11.71) = 12.

7. f(x) = [x]; x = 9.37 (under “floor” heading)
- Interpreting [x] as floor: floor(9.37) = 9.

8. f(x) = floor(x); x = 5.49
- floor(5.49) = 5.

9) Graph of the function f given by intervals
Intervals and constant values:
- 0 < x ≤ 1 : f(x) = 4
- 1 < x ≤ 2 : f(x) = 5
- 2 < x ≤ 3 : f(x) = 6
- 3 < x ≤ 4 : f(x) = 7
- 4 < x ≤ 5 : f(x) = 8
- 5 < x ≤ 6 : f(x) = 9

How to draw:
- On each interval (k, k+1] draw a horizontal segment at the indicated y-value.
- At each integer k = 1,2,3,4,5: place a closed dot at the right endpoint (x = k) of the segment to which it belongs, and an open dot at the left endpoint of the next segment (because the next segment excludes its left endpoint). Example: at x = 1 there is a closed dot at (1,4) and an open dot at (1,5).
- There is no point plotted at x = 0 (since 0 < x), and at x = 6 there is a closed dot at (6,9).

This produces a rising staircase of horizontal steps from y = 4 up to y = 9 on (0,6].