(score for question 3: ___ of 4 points)\n3. graph the step function over the interval -2 ≤ x ≤ 2.\nf(x)=⌈x +…

Explanation:

Step1: Analyze values at endpoints

When $x=-2$, $f(-2)=\lceil -2 + 3\rceil=\lceil1\rceil = 1$.

Step2: Analyze other integer - values in the interval

When $x=-1$, $f(-1)=\lceil -1 + 3\rceil=\lceil2\rceil = 2$. When $x = 0$, $f(0)=\lceil0 + 3\rceil=\lceil3\rceil = 3$. When $x = 1$, $f(1)=\lceil1+3\rceil=\lceil4\rceil = 4$. When $x = 2$, $f(2)=\lceil2 + 3\rceil=\lceil5\rceil = 5$. For $-2\leq x<-1$, $1\leq x + 3<2$, so $f(x)=\lceil x + 3\rceil = 2$. For $-1\leq x<0$, $2\leq x + 3<3$, so $f(x)=\lceil x + 3\rceil = 3$. For $0\leq x<1$, $3\leq x + 3<4$, so $f(x)=\lceil x + 3\rceil = 4$. For $1\leq x<2$, $4\leq x + 3<5$, so $f(x)=\lceil x + 3\rceil = 5$.

Step3: Graph the function

The graph will have a series of horizontal line - segments (steps). At $x=-2$, the function value is $1$. From $x=-2$ to $x = - 1$ (not including $x=-1$), the function value is $2$. From $x=-1$ to $x = 0$ (not including $x = 0$), the function value is $3$. From $x=0$ to $x = 1$ (not including $x = 1$), the function value is $4$. From $x=1$ to $x = 2$ (including $x = 2$), the function value is $5$.

Answer:

Graph has horizontal line - segments: at $y = 1$ for $x=-2$, at $y = 2$ for $-2<x<-1$, at $y = 3$ for $-1\leq x<0$, at $y = 4$ for $0\leq x<1$, at $y = 5$ for $1\leq x\leq2$.