Question

asked Jul 7, 2023 16.6k views

1 Answer

JamesRat · Answer 1 · 2023-07-12T07:19:52+0000

Answer:

y = -2 for -2 < x ≤ -1

Explanation:

Ceiling function

$f(x) = \left\lceil x \right\rceil$

The least integer that is greater than or equal to x.

Given ceiling function:

$f(x) =\left\lceil x-1 \right\rceil$

Evaluate values in the given intervals.

$\boxed{-4 \leq x < -3}$

$\implies f(-4) =\left\lceil -4-1 \right\rceil=\left\lceil -5 \right\rceil=-5$

$\implies f(-3.1) =\left\lceil -3.1-1 \right\rceil=\left\lceil -4.1 \right\rceil=-4$

$\boxed{-2 \leq x < -1}$

$\implies f(-2) =\left\lceil -2-1 \right\rceil=\left\lceil -3 \right\rceil=-3$

$\implies f(-1.1) =\left\lceil -1.1-1 \right\rceil=\left\lceil -2.1\right\rceil=-2$

$\boxed{-2 < x \leq -1}$

$\implies f(-1.9) =\left\lceil -1.9-1 \right\rceil=\left\lceil -2.9 \right\rceil=-2$

$\implies f(-1) =\left\lceil -1-1 \right\rceil=\left\lceil -2 \right\rceil=-2$

$\boxed{-4 < x \leq -3}$

$\implies f(-3.9) =\left\lceil -3.9-1 \right\rceil=\left\lceil -4.9 \right\rceil=-4$

$\implies f(-3) =\left\lceil -3-1 \right\rceil=\left\lceil -4\right\rceil=-4$

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).

From inspection of the attached graph of the given ceiling function:

This proves that the only step that is included in the graph of the given ceiling function from the given answer options is:

Please log in or register to add a comment.