1. **Problem Statement:** We are given a piecewise linear function defined on the interval from $x=0$ to $x=4$ with two segments: the first segment rises from $(0,0)$ to $(2,2)$, and the second segment is a horizontal line from $(2,2)$ to $(4,2)$. 2. **Formula and Explanation:** A piecewise linear function can be expressed as: $$ f(x) = \begin{cases} m_1 x + b_1 & \text{for } 0 \leq x \leq 2 \\ m_2 x + b_2 & \text{for } 2 < x \leq 4 \end{cases} $$ where $m_1$ and $m_2$ are slopes of the segments, and $b_1$, $b_2$ are the intercepts. 3. **First Segment (0 to 2):** - The line passes through $(0,0)$ and $(2,2)$. - Slope $m_1 = \frac{2 - 0}{2 - 0} = 1$. - Since it passes through the origin, $b_1 = 0$. - So, $f(x) = 1 \cdot x + 0 = x$ for $0 \leq x \leq 2$. 4. **Second Segment (2 to 4):** - The line is horizontal at $y=2$. - Slope $m_2 = 0$. - The function value is constant: $f(x) = 2$ for $2 < x \leq 4$. 5. **Final Piecewise Function:** $$ f(x) = \begin{cases} x & 0 \leq x \leq 2 \\ 2 & 2 < x \leq 4 \end{cases} $$ This function starts at the origin, increases linearly to $(2,2)$, then remains constant at $2$ from $x=2$ to $x=4$. **Answer:** The piecewise function is $f(x) = \begin{cases} x & 0 \leq x \leq 2 \\ 2 & 2 < x \leq 4 \end{cases}$.