Question

suppose that the function f is defined, for all real numbers, as follows.

f(x) = \

$$\begin{cases}2 & \\text{if }x < 1 \\\\1 & \\text{if }x = 1 \\\\3 & \\text{if }x > 1\\end{cases}$$

graph the function f.

Explanation:

Step1: Analyze \( x < 1 \) part

For \( x < 1 \), \( f(x)=2 \). This is a horizontal line segment (excluding \( x = 1 \)) at \( y = 2 \), from \( x=-\infty \) to \( x = 1 \) (with an open circle at \( x = 1 \) since \( x<1 \) does not include \( x = 1 \)).

Step2: Analyze \( x = 1 \) part

For \( x = 1 \), \( f(x)=1 \). This is a single point at \( (1,1) \) (a closed circle since \( x = 1 \) is included).

Step3: Analyze \( x > 1 \) part

For \( x > 1 \), \( f(x)=3 \). This is a horizontal line segment (excluding \( x = 1 \)) at \( y = 3 \), from \( x = 1 \) (open circle) to \( x=\infty \).

To graph:

• Draw a horizontal line at \( y = 2 \) for all \( x < 1 \), with an open circle at \( (1,2) \).
• Plot a closed circle at \( (1,1) \).
• Draw a horizontal line at \( y = 3 \) for all \( x > 1 \), with an open circle at \( (1,3) \).

(Note: Since the actual graphing tool is provided, the steps above guide the graphing process. The key is to represent each piece correctly with open/closed circles and horizontal lines.)

Answer:

The graph consists of:

• A horizontal segment at \( y = 2 \) for \( x < 1 \) (open circle at \( (1,2) \)),
• A closed dot at \( (1,1) \),
• A horizontal segment at \( y = 3 \) for \( x > 1 \) (open circle at \( (1,3) \)).