Question
Which point should have an open circle in the graph of the piecewise function
f(x) ={-x -1 if x <1 -x +2 if x > 1
A.(1,1)
B.(-1, -2)
C.(-1,1)
D.(1, -2)
f(x) ={-x -1 if x <1 -x +2 if x > 1
A.(1,1)
B.(-1, -2)
C.(-1,1)
D.(1, -2)
Answers
GPT-4o mini
To determine which point should have an open circle in the graph of the given piecewise function
\[ f(x) =
\begin{cases}
-x - 1 & \text{if } x < 1 \\
-x + 2 & \text{if } x > 1
\end{cases} \]
we need to analyze the value of the function at \( x = 1 \).
1. For \( x < 1 \), the function is defined by the first piece: \( f(x) = -x - 1 \). If we substitute \( x = 1 \) into this piece, we have:
\[
f(1) = -1 - 1 = -2.
\]
However, this piece is only valid for \( x < 1\), so \( f(1) \) is not included in this part of the function.
2. For \( x > 1 \), the second piece gives us \( f(x) = -x + 2 \). Substitute \( x = 1 \):
\[
f(1) = -1 + 2 = 1.
\]
Again, this piece only applies for \( x > 1\), so \( f(1) \) is not included here either.
Since the function \( f(x) \) is not defined for \( x = 1 \) (there’s no corresponding definition in either piece), we denote this point with an open circle.
The open circle at \( x = 1 \) indicates that the limit of the function approaches 1 as \( x \) approaches 1 from the right side but the function does not actually take on the value at \( f(1) \).
Based on the options provided:
- A. (1, 1) would represent the point where the function would be if it were defined; we want an open circle there since the function is undefined at that point.
- The other points do not seem related to the question about \( f(1) \).
Thus, the correct answer is:
**A. (1, 1)** (this point should have an open circle since \( f(1) \) is undefined).
\[ f(x) =
\begin{cases}
-x - 1 & \text{if } x < 1 \\
-x + 2 & \text{if } x > 1
\end{cases} \]
we need to analyze the value of the function at \( x = 1 \).
1. For \( x < 1 \), the function is defined by the first piece: \( f(x) = -x - 1 \). If we substitute \( x = 1 \) into this piece, we have:
\[
f(1) = -1 - 1 = -2.
\]
However, this piece is only valid for \( x < 1\), so \( f(1) \) is not included in this part of the function.
2. For \( x > 1 \), the second piece gives us \( f(x) = -x + 2 \). Substitute \( x = 1 \):
\[
f(1) = -1 + 2 = 1.
\]
Again, this piece only applies for \( x > 1\), so \( f(1) \) is not included here either.
Since the function \( f(x) \) is not defined for \( x = 1 \) (there’s no corresponding definition in either piece), we denote this point with an open circle.
The open circle at \( x = 1 \) indicates that the limit of the function approaches 1 as \( x \) approaches 1 from the right side but the function does not actually take on the value at \( f(1) \).
Based on the options provided:
- A. (1, 1) would represent the point where the function would be if it were defined; we want an open circle there since the function is undefined at that point.
- The other points do not seem related to the question about \( f(1) \).
Thus, the correct answer is:
**A. (1, 1)** (this point should have an open circle since \( f(1) \) is undefined).