Question

express the function graphed on the axes below as a piecewise function.
answer attempt 3 out of 3
$f(x)= \

$$\begin{cases} \\square & \\text{for } \\square \\\\ \\square & \\text{for } \\square \\end{cases}$$

$

Explanation:

Step1: Analyze the first segment (left part)

The first segment has a solid dot at \((-6, 8)\) and an open dot at \((-2, 6)\). Let's find the slope \(m\) between these two points. The slope formula is \(m=\frac{y_2 - y_1}{x_2 - x_1}\). So \(m=\frac{6 - 8}{-2 - (-6)}=\frac{-2}{4}=-\frac{1}{2}\). Using point - slope form \(y - y_1=m(x - x_1)\) with \((x_1,y_1)=(-6,8)\), we get \(y - 8=-\frac{1}{2}(x + 6)\), which simplifies to \(y=-\frac{1}{2}x+5\). The domain for this segment is \(x\leq - 2\) (since the left end is a solid dot at \(x = - 6\) and the right end is an open dot at \(x=-2\)).

Step2: Analyze the second segment (right part)

The second segment is a horizontal line with a \(y\) - value of \(6\). The left end is an open dot at \(x=-2\) and the right end is a solid dot at \(x = 3\). So the equation is \(y = 6\) and the domain is \(-2\lt x\leq3\).

Answer:

\[
f(x)=

$$\begin{cases} -\dfrac{1}{2}x + 5, & x\leq - 2\\ 6, & - 2\lt x\leq3 \end{cases}$$

\]