Domain and Range of Step Functions
The domain and range of step functions — specifically ceiling and floor functions — follow a clear pattern in enVision Algebra 1 Chapter 5 for Grade 11. Both the floor function f(x) = ⌊x⌋ and the ceiling function g(x) = ⌈x⌉ accept any real number as input, so their domain is (-∞, ∞). However, both always output integers: the floor rounds down and the ceiling rounds up. This means the range for both is the set of all integers {..., -2, -1, 0, 1, 2, ...}. For example, input x = 2.7 gives floor output 2 and ceiling output 3, both integers, which is why step function graphs appear as horizontal line segments at integer heights.
Key Concepts
For ceiling and floor functions: Domain: all real numbers, $( \infty, \infty)$ Range: all integers, $\{..., 2, 1, 0, 1, 2, ...\}$.
Common Questions
What is the domain of the floor function ⌊x⌋?
The domain is all real numbers, (-∞, ∞). Any real number can be input into the floor function.
What is the range of the ceiling function ⌈x⌉?
The range is all integers {..., -2, -1, 0, 1, 2, ...}. The ceiling function always rounds up to the nearest integer, so it only produces integer outputs.
Why does the floor function have integer outputs even for non-integer inputs?
Because the floor function rounds down to the nearest whole number. For example, ⌊2.7⌋ = 2 and ⌊-1.3⌋ = -2. Every output is an integer regardless of the input.
What is the difference between the floor and ceiling functions for the same input?
For a non-integer like 2.7, the floor gives 2 (rounds down) and the ceiling gives 3 (rounds up). For an integer like 3, both give the same value: ⌊3⌋ = ⌈3⌉ = 3.
Why do step function graphs show horizontal line segments at integer heights?
Because the output of floor and ceiling functions only changes at integer values of x, creating flat segments. The range being all integers means the graph only sits at integer y-values.