Answer:
y = -2 for -2 < x ≤ -1
Step-by-step explanation:
Ceiling function
[tex]f(x) = \left\lceil x \right\rceil[/tex]
The least integer that is greater than or equal to x.
Given ceiling function:
[tex]f(x) =\left\lceil x-1 \right\rceil[/tex]
Evaluate values in the given intervals.
[tex]\boxed{-4 \leq x < -3}[/tex]
[tex]\implies f(-4) =\left\lceil -4-1 \right\rceil=\left\lceil -5 \right\rceil=-5[/tex]
[tex]\implies f(-3.1) =\left\lceil -3.1-1 \right\rceil=\left\lceil -4.1 \right\rceil=-4[/tex]
[tex]\boxed{-2 \leq x < -1}[/tex]
[tex]\implies f(-2) =\left\lceil -2-1 \right\rceil=\left\lceil -3 \right\rceil=-3[/tex]
[tex]\implies f(-1.1) =\left\lceil -1.1-1 \right\rceil=\left\lceil -2.1\right\rceil=-2[/tex]
[tex]\boxed{-2 < x \leq -1}[/tex]
[tex]\implies f(-1.9) =\left\lceil -1.9-1 \right\rceil=\left\lceil -2.9 \right\rceil=-2[/tex]
[tex]\implies f(-1) =\left\lceil -1-1 \right\rceil=\left\lceil -2 \right\rceil=-2[/tex]
[tex]\boxed{-4 < x \leq -3}[/tex]
[tex]\implies f(-3.9) =\left\lceil -3.9-1 \right\rceil=\left\lceil -4.9 \right\rceil=-4[/tex]
[tex]\implies f(-3) =\left\lceil -3-1 \right\rceil=\left\lceil -4\right\rceil=-4[/tex]
Therefore, the step that is included in the graph of the given ceiling function is: y = -2 for -2 < x ≤ -1.
Graph of the parent ceiling function f(x) = ⌈x⌉
The graph of the parent function has open dots at (x, x+1) and closed dots at (x, x). Each segment is joined with a horizontal line (step).
The open dot means the value is not included, whereas the closed dot means the value is included.
To graph the given ceiling function f(x) = ⌈x - 1⌉
(See attachment).
- The y-value is 1 less than the parent function, therefore place open dots at (x, x) and closed dots at (x, x-1).
- Join each segment with a line (step).
From inspection of the attached graph of the given ceiling function:
- The step for y = -5 is -5 < x ≤ -4 (shown in blue).
- The step for y = -2 is -2 < x ≤ -1 (shown in red).
This proves that the only step that is included in the graph of the given ceiling function from the given answer options is:
