Absolute Value and Step Functions Homework Answer Key

1. Analyze $f(x)=|x+3|$ on $-6\le x\le2$

1(a) Evaluate the endpoints requested

$$f(-5)=|-5+3|=|-2|=2$$ $$f(2)=|2+3|=|5|=5$$

1(b) Complete the table and graph

Plot these points and connect them with straight segments. The graph is a V with vertex $(-3,0)$ and closed endpoints at $(-6,3)$ and $(2,5)$.

1(c) Increasing interval

The function decreases to the vertex at $x=-3$ and increases for $x>-3$. Of the choices, (2) $-2<x<1$ lies entirely on the increasing branch.

1(d) Range

The minimum value is $0$ at $x=-3$, and the largest value on the stated domain is $5$ at $x=2$. Therefore,

$$\boxed{0\le f(x)\le5}$$

2. Compare $|x-5|$ and $|x|-5$

They are not equivalent. One counterexample is enough: at $x=0$,

$$|0-5|=5,\qquad |0|-5=-5.$$

Because the outputs differ for the same input, the expressions cannot be equivalent for all $x$.

3. Graph the step functions

3(a)

For $-5\le x<0$, draw a horizontal segment at $y=-4$ with a closed point at $(-5,-4)$ and an open point at $(0,-4)$. For $0\le x\le5$, draw a horizontal segment at $y=4$ with closed points at $(0,4)$ and $(5,4)$.

3(b)

• Draw $y=10$ on $0\le x<4$: closed at $(0,10)$ and open at $(4,10)$.
• Draw $y=7$ on $4\le x<8$: closed at $(4,7)$ and open at $(8,7)$.
• Draw $y=4$ on $8\le x\le12$: closed at both $(8,4)$ and $(12,4)$.

At a boundary such as $x=4$, the open point on the old step and the closed point on the new step show which output the function actually takes.

4. Postage step function

4(a) Letter weighing 1.5 ounces

The interval $1\le x<2$ has a charge of 0.70 dollars, so a 1.5-ounce letter costs 0.70 dollars.

4(b) Letter weighing exactly 3.0 ounces

At $x=3$, the 0.90-dollar step has an open point and the 1.15-dollar step has a closed point. The charge is therefore 1.15 dollars.

4(c) Piecewise definition

The graph can be written as

$$p(x)= \begin{cases} 0.50, & 0<x<1,\\ 0.70, & 1\le x<2,\\ 0.90, & 2\le x<3,\\ 1.15, & 3\le x<4. \end{cases}$$

4(d) Why $0.50\le y\le1.15$ is not the range

That interval notation would include every value between 0.50 and 1.15 dollars, but the step function outputs only four charges. Its range is

$$\boxed{\{0.50,\ 0.70,\ 0.90,\ 1.15\}}.$$