\( y = \left\{ \begin{array} { l } { \frac { 4 } { x } , x < - 2 } \\ { x , - 2 \leq x < 0 } \\ { 1 - x , x \geq 0 } \end{array} \right. \)

This piecewise function has three distinct segments that beautifully illustrate how functions can behave differently based on the value of \( x \). For \( x < -2 \), the function takes the form \( \frac{4}{x} \), creating a hyperbolic curve that approaches zero as \( x \) decreases. In the second segment, where \( -2 \leq x < 0 \), it simply mirrors the input as \( y = x \), a straight line portion. Finally, for \( x \geq 0 \), the function transitions to a linear equation \( y = 1 - x \), indicating a line descending from the point (0,1). Graphing this function can truly bring its behavior to life! Start by plotting the hyperbola for values less than -2, a diagonal line from (-2,-2) to (0,0), and then a descending line starting at (0,1) downwards. Each segment connects nicely, presenting an engaging visual representation of how a single function can exhibit different characteristics based on \( x \). Don’t forget to mark those critical points where the function changes definitions; they often become essential points of interest in analysis!