\( \left. \begin{array} { c } { f ( x ) = \left\{ \begin{array} { l l } { x - 1 , } & { x \leq 1 } \\ { 6 - 2 x , } & { 1 < x < 3 } \\ { x + 1 , } & { x \geq 3 } \end{array} \right.} \\ { g ( x ) = \left\{ \begin{array} { l l } { x + 1 , } & { x < 1 } \\ { x - 1 , } & { x \geq 1 } \end{array} \right.} \end{array} \right. \)

We have two piecewise functions defined as follows: 1. The function \( f(x) \): \[ f(x) = \begin{cases} x - 1 & \text{if } x \leq 1 \\ 6 - 2x & \text{if } 1 < x < 3 \\ x + 1 & \text{if } x \geq 3 \end{cases} \] 2. The function \( g(x) \): \[ g(x) = \begin{cases} x + 1 & \text{if } x < 1 \\ x - 1 & \text{if } x \geq 1 \end{cases} \] ### Analyzing the Functions #### Function \( f(x) \) - For \( x \leq 1 \): \[ f(x) = x - 1 \] This is a linear function with a slope of 1. - For \( 1 < x < 3 \): \[ f(x) = 6 - 2x \] This is also a linear function, but with a slope of -2. - For \( x \geq 3 \): \[ f(x) = x + 1 \] This is again a linear function with a slope of 1. #### Function \( g(x) \) - For \( x < 1 \): \[ g(x) = x + 1 \] This is a linear function with a slope of 1. - For \( x \geq 1 \): \[ g(x) = x - 1 \] This is also a linear function with a slope of 1. ### Key Points and Values To understand the behavior of these functions, we can evaluate them at key points, particularly at the boundaries of the piecewise definitions. 1. **At \( x = 1 \)**: - \( f(1) = 1 - 1 = 0 \) - \( g(1) = 1 - 1 = 0 \) 2. **At \( x = 3 \)**: - \( f(3) = 3 + 1 = 4 \) - \( g(3) = 3 - 1 = 2 \) 3. **At \( x = 0 \)**: - \( f(0) = 0 - 1 = -1 \) - \( g(0) = 0 + 1 = 1 \) 4. **At \( x = 2 \)**: - \( f(2) = 6 - 2(2) = 2 \) - \( g(2) = 2 - 1 = 1 \) ### Summary of Values - \( f(0) = -1 \), \( g(0) = 1 \) - \( f(1) = 0 \), \( g(1) = 0 \) - \( f(2) = 2 \), \( g(2) = 1 \) - \( f(3) = 4 \), \( g(3) = 2 \) ### Graphing the Functions To visualize these functions, you can plot them on a graph. The function \( f(x) \) will have three segments, while \( g(x) \) will have two segments. The slopes and intercepts will help you sketch the lines accurately. If you have any specific questions about these functions or need further analysis, feel free to ask!