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 first segment (left part)

The left part of the graph passes through the origin \((0,0)\) and has a point \((1,6)\). The slope \(m\) is calculated as \(\frac{y_2 - y_1}{x_2 - x_1}=\frac{6 - 0}{1 - 0}=6\). So the equation is \(y = 6x\). The domain here: the solid dot is at \(x = 1\), and the line extends left, so for \(x\leq1\).

Step2: Analyze the second segment (right part)

The right part starts with an open circle at \((4,8)\) and has a positive slope. Let's find the slope between \((4,8)\) and another point, say if we take a point like \((5,9)\) (since it's a line with slope 1, as \(9 - 8=1\) and \(5 - 4 = 1\)). So the equation: using point - slope form \(y - y_1=m(x - x_1)\), with \(m = 1\), \(x_1 = 4\), \(y_1 = 8\), we get \(y-8=1\times(x - 4)\), so \(y=x + 4\). The domain here: the open circle at \(x = 4\), so for \(x>4\).

Answer:

$$ f(x)=
LATEXBLOCK0
$$