Question

express the function graphed on the axes below as a piecewise function.
answer attempt 1 out of 3
$f(x) = \

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

$

Explanation:

Step1: Analyze the left line

The left line passes through \((-8, 0)\) and \((-4, 5)\). The slope \(m_1=\frac{5 - 0}{-4 - (-8)}=\frac{5}{4}\). Using point - slope form \(y - y_1=m(x - x_1)\) with \((x_1,y_1)=(-8,0)\), we get \(y-0=\frac{5}{4}(x + 8)\), which simplifies to \(y=\frac{5}{4}x + 10\). The domain of this line is \(x\lt - 4\) (since there is an open circle at \(x=-4\)).

Step2: Analyze the right line

The right line passes through \((2, 6)\) and \((8, 0)\). The slope \(m_2=\frac{0 - 6}{8 - 2}=\frac{-6}{6}=-1\). Using point - slope form \(y - y_1=m(x - x_1)\) with \((x_1,y_1)=(2,6)\), we get \(y - 6=-1(x - 2)\), which simplifies to \(y=-x + 8\). The domain of this line is \(x\gt2\) (since there is an open circle at \(x = 2\)).

Answer:

\(f(x)=

$$\begin{cases}\frac{5}{4}x + 10& \text{for }x\lt - 4\\-x + 8& \text{for }x\gt2\end{cases}$$

\)