Graph the function. \[ f(x)=\left\{\begin{array}{ll}|x| & \text { for } x<1 \\ -x+2 & \text { for } x \geq 1\end{array}\right. \]

It seems that I am unable to generate a graph directly. However, I can guide you through the process of graphing the piecewise function step by step. The function is defined as follows: \[ f(x)=\left\{ \begin{array}{ll} |x| & \text{for } x<1 \\ -x+2 & \text{for } x \geq 1 \end{array} \right. \] ### Step 1: Graph the first piece \( f(x) = |x| \) for \( x < 1 \) 1. **Identify the function**: The absolute value function \( |x| \) is a V-shaped graph that opens upwards. 2. **Domain**: This part of the function is valid for \( x < 1 \). 3. **Key points**: - At \( x = 0 \), \( f(0) = |0| = 0 \). - As \( x \) approaches 1 from the left, \( f(1) = |1| = 1 \) (but this point is not included since it is for \( x < 1 \)). 4. **Plot the line**: Draw the line segment from \( (-\infty, 0) \) to \( (0, 0) \) and from \( (0, 0) \) to just before \( (1, 1) \). ### Step 2: Graph the second piece \( f(x) = -x + 2 \) for \( x \geq 1 \) 1. **Identify the function**: This is a linear function with a negative slope. 2. **Domain**: This part of the function is valid for \( x \geq 1 \). 3. **Key points**: - At \( x = 1 \), \( f(1) = -1 + 2 = 1 \) (this point is included). - At \( x = 2 \), \( f(2) = -2 + 2 = 0 \). 4. **Plot the line**: Draw the line starting from \( (1, 1) \) and continuing downwards to the right. ### Step 3: Combine the two pieces - The graph will consist of two parts: - The first part is a V-shape from the left up to but not including the point \( (1, 1) \). - The second part is a straight line starting from \( (1, 1) \) and going downwards. ### Final Notes - Make sure to indicate that the point at \( (1, 1) \) is included in the second piece but not in the first. - You can use graphing software or a graphing calculator to visualize this piecewise function accurately.