Question

which points lie on the graph of the function $f(x)=\lceil x \
ceil + 2$? check all that apply.\
$\square$ $(-5.5, -4)$\
$\square$ $(-3.8, -2)$\
$\square$ $(-1.1, 1)$\
$\square$ $(-0.9, 2)$\
$\square$ $(2.2, 5)$\
$\square$ $(4.7, 6)$

Explanation:

Step1: Recall the ceiling function

The ceiling function $\lceil x
ceil$ gives the smallest integer greater than or equal to $x$. We will apply this to each $x$-value in the points and check if $\lceil x
ceil + 2$ equals the $y$-value.

Step2: Check $(-5.5, -4)$

For $x = -5.5$, $\lceil -5.5
ceil=-5$ (since -5 is the smallest integer $\geq -5.5$). Then $\lceil -5.5
ceil + 2=-5 + 2=-3
eq -4$. So this point does not lie on the graph.

Step3: Check $(-3.8, -2)$

For $x=-3.8$, $\lceil -3.8
ceil=-3$ (smallest integer $\geq -3.8$). Then $\lceil -3.8
ceil + 2=-3 + 2=-1
eq -2$. So this point does not lie on the graph.

Step4: Check $(-1.1, 1)$

For $x = -1.1$, $\lceil -1.1
ceil=-1$ (smallest integer $\geq -1.1$). Then $\lceil -1.1
ceil + 2=-1 + 2 = 1$. So this point lies on the graph.

Step5: Check $(-0.9, 2)$

For $x=-0.9$, $\lceil -0.9
ceil=0$ (smallest integer $\geq -0.9$). Then $\lceil -0.9
ceil + 2=0 + 2 = 2$. So this point lies on the graph.

Step6: Check $(2.2, 5)$

For $x = 2.2$, $\lceil 2.2
ceil=3$ (smallest integer $\geq 2.2$). Then $\lceil 2.2
ceil + 2=3 + 2 = 5$. So this point lies on the graph.

Step7: Check $(4.7, 6)$

For $x = 4.7$, $\lceil 4.7
ceil=5$ (smallest integer $\geq 4.7$). Then $\lceil 4.7
ceil + 2=5 + 2 = 7
eq 6$. So this point does not lie on the graph.

Answer:

$(-1.1, 1)$, $(-0.9, 2)$, $(2.2, 5)$