Determine if the endpoints for the following piecewise function are opened or closed

Question

-3 x < 2
{2 x ≤ x < 4
5 x ≥ 4

Which of the following answer are open or closed

4,2
4,5
2,2
2, -3

Answer 1

To determine whether the endpoints for the given piecewise function are open or closed, we can analyze each segment of the function.

The piecewise function can be divided as follows:

1. For \( x < 2 \): The endpoint at 2 is excluded, which means it is open.
2. For \( 2 \leq x < 4 \): The endpoint at 2 is included (because of the \( \leq \)), so it is closed; however, the endpoint at 4 is excluded (since the interval is \( x < 4 \)), so it is open.
3. For \( x \geq 4 \): The endpoint at 4 is included (because of the \( \geq \)), so it is closed.

Now let's evaluate the endpoints you've provided:

1. 4, 2:

   • 4: Closed (included in \( x \geq 4 \)).
   • 2: Closed (included in \( 2 \leq x < 4 \)).
   • Result: Closed.

2. 4, 5:

   • 4: Closed (included in \( x \geq 4 \)).
   • 5: Closed (included in \( x \geq 4 \)).
   • Result: Closed.

3. 2, 2:

   • 2: Closed (included in \( 2 \leq x < 4 \)).
   • 2: Closed (included in \( 2 \leq x < 4 \)).
   • Result: Closed.

4. 2, -3:

   • 2: Closed (included in \( 2 \leq x < 4 \)).
   • -3: Open (since we have \( x < 2 \), -3 is not included).
   • Result: Mixed (Closed, Open).

To summarize:

• 4, 2: Closed
• 4, 5: Closed
• 2, 2: Closed
• 2, -3: Mixed (Closed, Open)