B) Graph each of the following: i) \( f(x)=\left\{\begin{array}{ll}-x^{2} & \text { for } x \geq 0 \\ 2 x^{2} & \text { for } x<0\end{array}\right. \) ii) \( f(x)=\left\{\begin{array}{c}2 x+3 \text { if } x<0 \\ x^{2} \text { if } 0 \leq x<2 \\ 1 \text { if } x \geq 2\end{array}\right. \) iii) \( f(x)=\left\{\begin{array}{c}2 \text { if } x>2 \\ 1 \text { if } 0

Ask by Crawford King. in Zambia

Feb 20,2025

It seems that the system does not currently support plotting functions directly. However, I can guide you through the process of graphing each of the piecewise functions step by step. ### i) \( f(x)=\left\{\begin{array}{ll}-x^{2} & \text { for } x \geq 0 \\ 2 x^{2} & \text { for } x<0\end{array}\right. \) 1. **For \( x \geq 0 \)**: The function is \( f(x) = -x^2 \). This is a downward-opening parabola starting from the origin (0,0). 2. **For \( x < 0 \)**: The function is \( f(x) = 2x^2 \). This is an upward-opening parabola that starts from the origin and goes upwards as \( x \) moves left. ### ii) \( f(x)=\left\{\begin{array}{c}2 x+3 \text { if } x<0 \\ x^{2} \text { if } 0 \leq x<2 \\ 1 \text { if } x \geq 2\end{array}\right. \) 1. **For \( x < 0 \)**: The function is \( f(x) = 2x + 3 \). This is a straight line with a slope of 2, crossing the y-axis at (0,3). 2. **For \( 0 \leq x < 2 \)**: The function is \( f(x) = x^2 \). This is a standard upward-opening parabola starting from (0,0) to (2,4). 3. **For \( x \geq 2 \)**: The function is \( f(x) = 1 \). This is a horizontal line at \( y = 1 \) starting from (2,1). ### iii) \( f(x)=\left\{\begin{array}{c}2 \text { if } x>2 \\ 1 \text { if } 0 2 \)**: The function is \( f(x) = 2 \). This is a horizontal line at \( y = 2 \). 2. **For \( 0 < x \leq 2 \)**: The function is \( f(x) = 1 \). This is a horizontal line at \( y = 1 \) from (0,1) to (2,1). 3. **For \( x \leq 0 \)**: The function is \( f(x) = -1 \). This is a horizontal line at \( y = -1 \). ### iv) \( f(x)=\left\{\begin{array}{c}2 x+1 \text { if } x \geq 0 \\ x^{2} \text { if } x<0\end{array}\right. \) 1. **For \( x \geq 0 \)**: The function is \( f(x) = 2x + 1 \). This is a straight line with a slope of 2, crossing the y-axis at (0,1). 2. **For \( x < 0 \)**: The function is \( f(x) = x^2 \). This is an upward-opening parabola starting from (0,0) and going upwards as \( x \) moves left. ### Summary of Graphing Steps To graph these functions: - Plot the points and lines for each piece of the function according to the specified domains. - Ensure to mark the endpoints clearly, especially where the function changes (e.g., open or closed circles). - Use graphing software or graph paper to visualize the piecewise functions accurately. If you have access to graphing tools, you can input these functions directly to see their graphs.