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

To graph the piecewise function \( f(x) \), we need to consider its two different rules based on the value of \( x \). 1. For \( x < 1 \): The function is \( f(x) = |x| \). This means for negative values of \( x \), the graph is a line whose value is the positive counterpart of \( x \). For example, at \( x = -1 \), \( f(-1) = 1 \), and at \( x = 0 \), \( f(0) = 0 \). 2. For \( x \geq 1 \): The function is \( f(x) = -x + 2 \). This is a straight line with a negative slope. At \( x = 1 \), \( f(1) = 1 \), and at \( x = 2 \), \( f(2) = 0 \). To graph: - Start by plotting the part \( f(x) = |x| \) for \( x < 1 \). This portion of the graph is the V-shape of the absolute value. It rises to the point \( (1, 1) \), but note that this point is open (not included since it's less than 1). - At \( x = 1 \), draw the point \( (1, 1) \) solid since it is included in the second part of the function. - For \( x \geq 1 \), graph the line \( f(x) = -x + 2 \) starting from \( (1, 1) \) and continuing downwards. This line will cross the x-axis when \( f(x) = 0 \), specifically at \( x = 2 \). The complete graph looks like a V rising on the left and a straight line falling on the right, meeting smoothly at the point \( (1, 1) \). By combining both parts, you create a unique piecewise graph reflecting both behaviors of the function.