Question

graph the piecewise function given below.
$f(x)=\

$$\begin{cases}-x & \\text{for } -3 < x < 4 \\\\ |x - 5| + 4 & \\text{for } 4 < x < 7\\end{cases}$$

$
step 2: drag the double blue arrows inward if necessary to set the domain of the function. afterwards you may click on the closed circle to create an open circle. done

Explanation:

Step1: Define first piece domain

Domain: $-3 < x < 4$

Step2: Graph first linear piece

Function: $f(x) = -x$

• At $x=-3$: $f(-3) = 3$ (open circle, since $x>-3$)
• At $x=4$: $f(4) = -4$ (open circle, since $x<4$)
• Plot line through points $(-3,3)$ (open) and $(4,-4)$ (open)

Step3: Define second piece domain

Domain: $4 < x < 7$

Step4: Graph second absolute value piece

Function: $f(x) = |x-5| + 4$

• Vertex at $x=5$: $f(5) = |5-5| +4 = 4$ (point $(5,4)$)
• At $x=4$: $f(4) = |4-5| +4 = 5$ (open circle, since $x>4$)
• At $x=7$: $f(7) = |7-5| +4 = 6$ (open circle, since $x<7$)
• Plot V-shaped graph through $(4,5)$ (open), $(5,4)$, $(7,6)$ (open)

Answer:

1. For $-3 < x < 4$: A straight line with open circles at $(-3, 3)$ and $(4, -4)$, passing through the origin $(0,0)$.
2. For $4 < x < 7$: A V-shaped graph with open circles at $(4, 5)$ and $(7, 6)$, with the vertex at $(5, 4)$.