In the graph of the piecewise function f(x)={2x if x≥1−14x−3 if x<0

Question

, which piece would have an open endpoint?(1 point)
Responses

neither function
neither function

f(x)=−14x−3
f left parenthesis x right parenthesis equals negative Start Fraction 1 over 4 End Fraction x minus 3

f(x)=2x
f left parenthesis x right parenthesis equals 2 x

both functions

Answer 1

To determine which piece of the piecewise function has an open endpoint, we can look at the definitions of the function in the intervals defined by the piecewise function.

The piecewise function is defined as follows:

• For \( x \geq 1 \), \( f(x) = 2x \)
• For \( x < 0 \), \( f(x) = -\frac{1}{4}x - 3 \)

Now, we need to examine the endpoints for these pieces:

1. For the piece \( f(x) = 2x \) (which is applicable for \( x \geq 1 \)), the endpoint at \( x = 1 \) is included (closed) in this interval because it uses the "greater than or equal to" condition.

2. For the piece \( f(x) = -\frac{1}{4}x - 3 \) (which is applicable for \( x < 0 \)), this function does not include the point at \( x = 0 \), which is where this piece transitions to the other piece. Hence, the endpoint at \( x = 0 \) is open.

Therefore, the piece that has an open endpoint is:

f(x) = -\frac{1}{4}x - 3 (for \( x < 0 \)).