Question
In the graph of the piecewise function f(x) =2x if x > 1
-1/4x -3 if x <0,
Which piece would have an open endpoint?
A: f(x) =2x
both functions
neither function
f(x) = -1/4x -3
-1/4x -3 if x <0,
Which piece would have an open endpoint?
A: f(x) =2x
both functions
neither function
f(x) = -1/4x -3
Answers
GPT-4o mini
To determine which piece of the piecewise function has an open endpoint, we need to analyze the conditions for each piece:
1. For \( f(x) = 2x \), the condition is \( x > 1 \). This means at the point \( x = 1 \), the function does not include this endpoint; instead, it is an open endpoint. So, for this piece, \( f(x) \) is defined for all \( x \) greater than 1 but not equal to 1.
2. For \( f(x) = -\frac{1}{4}x - 3 \), the condition is \( x < 0 \). This means that the piece does not define the function for \( x = 0 \) (the endpoint is open as well). So for this piece, \( f(x) \) is defined for all \( x \) less than 0 but not equal to 0.
Since both pieces have open endpoints, the answer is **both functions**.
1. For \( f(x) = 2x \), the condition is \( x > 1 \). This means at the point \( x = 1 \), the function does not include this endpoint; instead, it is an open endpoint. So, for this piece, \( f(x) \) is defined for all \( x \) greater than 1 but not equal to 1.
2. For \( f(x) = -\frac{1}{4}x - 3 \), the condition is \( x < 0 \). This means that the piece does not define the function for \( x = 0 \) (the endpoint is open as well). So for this piece, \( f(x) \) is defined for all \( x \) less than 0 but not equal to 0.
Since both pieces have open endpoints, the answer is **both functions**.