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

Question

2x x < 2
f(x) = {3x x ≤ x < 4
4x x > 55

What would be the last point graphed on the second piece of the function, f (x) = 3x

Answer 1

To determine whether the endpoints of the piecewise function

$$f(x)=\{\begin{array}{llll}2x& \text{if }x<23x& \text{if }2\le x<44x& \text{if }x>5\end{array}$$

are open or closed, we need to analyze the conditions at the transition points of the piecewise function.

1. For the first piece: $f(x)=2x$ for $x<2$

   • The endpoint $x=2$ is not included in this piece, which means it is open.

2. For the second piece: $f(x)=3x$ for $2\le x<4$

   • Here, $x=2$ is included (closed), and $x=4$ is not included (open).

3. For the third piece: $f(x)=4x$ for $x>5$

   • The endpoint $x=5$ is not included in this piece, meaning it is open.

Now, to answer the question about the last point graphed on the second piece of the function $f(x)=3x$:

• The last point of this piece would occur at $x=4$. Since the condition for this piece is $2\le x<4$, the function does not include $x=4$. Therefore, the last point graphed on this segment is at $x=4$ (open).

In conclusion:

• At $x=2$: Closed
• At $x=4$: Open
• At $x=5$: Open

The answer to your specific question is that the last point graphed on the second piece $f(x)=3x$ is at $x=4$ (open).