Question

graph the function.
$f(x) = \

$$\begin{cases} x + 1 & \\text{if } x < 1 \\\\ 3 & \\text{if } x \\geq 1 \\end{cases}$$

$

Explanation:

Step1: Analyze the first piece ($x < 1$)

The function is $f(x)=x + 1$ for $x < 1$. This is a linear function with slope $1$ and y - intercept $1$. We can find some points: when $x = 0$, $f(0)=0 + 1=1$; when $x=-1$, $f(-1)=-1 + 1 = 0$. Since $x < 1$, the point at $x = 1$ for this piece is not included (open circle at $(1,1 + 1)=(1,2)$).

Step2: Analyze the second piece ($x\geq1$)

The function is $f(x)=3$ for $x\geq1$. This is a horizontal line. At $x = 1$, $f(1)=3$ (closed circle at $(1,3)$), and for $x>1$, like $x = 2$, $f(2)=3$, $x = 3$, $f(3)=3$, etc.

Step3: Graphing steps

1. For the line $y=x + 1$ (where $x < 1$): Plot the points we found (e.g., $(-1,0),(0,1)$) and draw a line with slope $1$, ending with an open circle at $(1,2)$.
2. For the horizontal line $y = 3$ (where $x\geq1$): Plot a closed circle at $(1,3)$ and draw a horizontal line to the right from this point.

(Note: Since the question asks to graph the function, the final answer is the graphical representation as described above. If we were to describe the graph: The left part (for $x < 1$) is a line with slope 1 passing through (0,1) with an open circle at (1,2). The right part (for $x\geq1$) is a horizontal line $y = 3$ starting with a closed circle at (1,3) and extending to the right.)

Answer:

The graph consists of two parts: a line segment of \( y = x + 1 \) for \( x < 1 \) (with an open circle at \( (1, 2) \)) and a horizontal line \( y = 3 \) for \( x\geq1 \) (with a closed circle at \( (1, 3) \)).