Question

express the function graphed on the axes below as a piecewise function.
answer attempt 2 out of 2
$f(x) = \

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

$

Explanation:

Step1: Analyze the first piece

The left - most part of the graph is a horizontal line. A horizontal line has a slope of 0. The y - value of this horizontal line is 4. The domain for this part: looking at the graph, the open circle is at \(x = 2\) (wait, no, looking at the graph, the first segment: the horizontal line goes from \(x=-\infty\) up to \(x = 2\) (the open circle at \(x = 2,y = 4\))? Wait, no, the first open circle is at \(x = 2\) (since the x - axis grid: from the origin, moving to \(x = 2\)). Wait, the first segment: horizontal line \(y = 4\), with an open circle at \(x=2\), so the domain is \(x<2\). So the first function is \(f(x)=4\) for \(x < 2\).

Step2: Analyze the second piece

The second segment is a line. Let's find the equation of the line. We have two points: the open circle at \((5,3)\) and the point \((8,0)\). First, calculate the slope \(m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{0 - 3}{8 - 5}=\frac{- 3}{3}=- 1\). Using the point - slope form \(y - y_1=m(x - x_1)\), using the point \((8,0)\): \(y-0=-1(x - 8)\), so \(y=-x + 8\). The domain for this part: the open circle is at \(x = 5\) and it goes to \(x>5\) (wait, no, the open circle is at \(x = 5,y = 3\) and the line goes to \(x = 8\) (where it touches the x - axis) and beyond? Wait, no, the open circle is at \(x = 5\), so the domain is \(x>5\)? Wait, no, the first segment is \(x<2\), the second segment starts at \(x>5\)? Wait, maybe I made a mistake. Wait, the first open circle is at \(x = 2\) (y = 4), the second open circle is at \(x = 5\) (y = 3). So between \(x = 2\) and \(x = 5\), is there a gap? Wait, the graph: the first part is horizontal from \(x=-\infty\) to \(x = 2\) (open circle at \(x = 2\)), then the second part is a line starting at \(x = 5\) (open circle at \(x = 5,y = 3\)) and going to \(x=\infty\) with a slope of - 1. Wait, let's re - examine the graph.

Wait, the first segment: horizontal line \(y = 4\), with an open circle at \(x = 2\) (so \(x<2\)). The second segment: a line with open circle at \(x = 5\) (y = 3) and passing through \((8,0)\). Let's confirm the slope between \((5,3)\) and \((8,0)\): \(m=\frac{0 - 3}{8 - 5}=-1\). Equation: \(y=3-1(x - 5)=-x + 8\). So the domain for the second function is \(x>5\). So the piecewise function is:

\(f(x)=

$$\begin{cases}4, & x < 2\\-x + 8, & x>5\end{cases}$$

\)

Answer:

\(f(x)=

$$\begin{cases}4 & \text{for } x < 2\\-x + 8 & \text{for } x>5\end{cases}$$

\)