12) Graph the function $$ f(x)=\left\{\begin{array}{cc}1 ext { if } x0 \ x+1, & ext { if } x \leq 0\end{array} ight. $$

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

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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.
To graph the function $$ f(x) $$, draw a horizontal line at $$ y = 1 $$ for $$ x > 0 $$ and a line starting at (0, 1) with a slope of 1 for $$ x \leq 0 $$. The point (0, 1) is included as a solid dot.

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

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