given the function below, fill in the table of values, use the table of values to graph the function, and…

Explanation:

Step1: Calculate $x-5$ for each $x$

For $x=2$: $2-5=-3$ For $x=3$: $3-5=-2$ For $x=4$: $4-5=-1$ For $x=5$: $5-5=0$ For $x=6$: $6-5=1$ For $x=7$: $7-5=2$ For $x=8$: $8-5=3$

Step2: Calculate $|x-5|$ for each result

For $x=2$: $|-3|=3$ For $x=3$: $|-2|=2$ For $x=4$: $|-1|=1$ For $x=5$: $|0|=0$ For $x=6$: $|1|=1$ For $x=7$: $|2|=2$ For $x=8$: $|3|=3$

Step3: Calculate $y=\frac{1}{2}|x-5|$

For $x=2$: $\frac{1}{2} \times 3 = \frac{3}{2}$ For $x=3$: $\frac{1}{2} \times 2 = 1$ For $x=4$: $\frac{1}{2} \times 1 = \frac{1}{2}$ For $x=5$: $\frac{1}{2} \times 0 = 0$ For $x=6$: $\frac{1}{2} \times 1 = \frac{1}{2}$ For $x=7$: $\frac{1}{2} \times 2 = 1$ For $x=8$: $\frac{1}{2} \times 3 = \frac{3}{2}$

Step4: Identify domain and range

Domain: All real numbers (no restrictions on $x$ for absolute value functions). Range: All non-negative real numbers (absolute value outputs $\geq0$, scaled by positive $\frac{1}{2}$ so $y\geq0$).

Answer:

Completed Table:

| $x$ | $x-5$ | $|x-5|$ | $y=\frac{1}{2}|x-5|$ | |-----|-------|---------|-----------------------| | 2 | $-3$ | $3$ | $\frac{3}{2}$ | | 3 | $-2$ | $2$ | $1$ | | 4 | $-1$ | $1$ | $\frac{1}{2}$ | | 5 | $0$ | $0$ | $0$ | | 6 | $1$ | $1$ | $\frac{1}{2}$ | | 7 | $2$ | $2$ | $1$ | | 8 | $3$ | $3$ | $\frac{3}{2}$ |

Domain and Range:

Domain: All real numbers, or $(-\infty, \infty)$ Range: All non-negative real numbers, or $[0, \infty)$