Question

e function graphed on the axes below as a piecewise functio
attempt 2 out of 2
= \

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

Explanation:

Step1: Analyze the first piece

The horizontal line segment for \( x < 2 \) has a \( y \)-value of 4. So the first part of the piecewise function is \( 4 \) when \( x < 2 \).

Step2: Analyze the second piece

For the line segment where \( 2 < x < 5 \), we find the slope and equation. The line passes through points (5, 3) and (8, 0) (but we use the interval \( 2 < x < 5 \)). Using two points on this line, say (5, 3) and (8, 0), the slope \( m=\frac{0 - 3}{8 - 5}=- 1 \). Using point - slope form \( y - y_1=m(x - x_1) \) with (5, 3), we get \( y-3=-1(x - 5)\), which simplifies to \( y=-x + 8 \). So for \( 2 < x < 5 \), the function is \( -x + 8 \).

Answer:

The piecewise function is \( f(x)=

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

\)