12) Graph the function \( f(x)=\left\{\begin{array}{cc}1 \text { if } x>0 \\ x+1, & \text { if } x \leq 0\end{array}\right. \)

To graph the function \[ f(x) = \begin{cases} 1 & \text{if } x > 0 \\ x + 1 & \text{if } x \leq 0 \end{cases} \] we will analyze each piece of the function separately. 1. **For \( x > 0 \)**: - The function is constant and equal to 1. This means that for any positive value of \( x \), the output \( f(x) \) will be 1. - On the graph, this will be a horizontal line at \( y = 1 \) starting from \( x = 0 \) and extending to the right. 2. **For \( x \leq 0 \)**: - The function is defined as \( f(x) = x + 1 \). This is a linear function with a slope of 1 and a y-intercept of 1. - To find specific points: - At \( x = 0 \): \( f(0) = 0 + 1 = 1 \) - At \( x = -1 \): \( f(-1) = -1 + 1 = 0 \) - At \( x = -2 \): \( f(-2) = -2 + 1 = -1 \) - This line will start at the point (0, 1) and will decrease as \( x \) becomes more negative. ### Steps to Graph: - Draw a horizontal line at \( y = 1 \) for \( x > 0 \). - For \( x \leq 0 \), plot the line segment starting from (0, 1) and going through the points (-1, 0) and (-2, -1). - The line for \( x \leq 0 \) will continue indefinitely to the left. ### Important Points: - The point (0, 1) is included in both pieces of the function, so it will be a solid dot. - The line for \( x + 1 \) will not be defined for \( x > 0 \). This gives you a complete picture of the function's graph. If you have graphing software or graph paper, you can plot these points and lines accordingly.