Question

In a graph of the function f(x)=⎧⎩⎨⎪⎪12x+2 if x<−3−6 if −3≤x<3−x+2 if x≥3
, which points will have open circles?(1 point)
Responses

(3,−1)
and (3,−6)
left parenthesis 3 comma negative 1 right parenthesis and left parenthesis 3 comma negative 6 right parenthesis

none of the points
none of the points

(−3,12)
, (3,−1)
, and (3,−6)
left parenthesis negative 3 comma Start Fraction 1 over 2 End Fraction right parenthesis , left parenthesis 3 comma negative 1 right parenthesis , and left parenthesis 3 comma negative 6 right parenthesis

(−3,12)
and (3,−6)

Answer 1

To determine which points will have open circles in the graph of the piecewise function

\[
f(x) =
\begin{cases}
\frac{1}{2}x + 2 & \text{if } x < -3 \\
-6 & \text{if } -3 \leq x < 3 \\
-x + 2 & \text{if } x \geq 3
\end{cases}
\]

we need to analyze the function at the points where the function changes.

1. **At \(x = -3\)**:
- For \(x < -3\), the function is given by \(\frac{1}{2}x + 2\).
If we substitute \(x = -3\) into the equation:
\[ f(-3) = \frac{1}{2}(-3) + 2 = -\frac{3}{2} + 2 = \frac{1}{2} \]
- For \(-3 \leq x < 3\), the function value is \(f(x) = -6\).
- Thus, at \(x = -3\), it transitions from \(f(-3) = \frac{1}{2}\) to \(f(-3) = -6\). This means there will be an open circle at \((-3, \frac{1}{2})\).

2. **At \(x = 3\)**:
- For \(x < 3\), \(f(x) = -6\) (constant for \(-3 \leq x < 3\)).
- For \(x \geq 3\), \(f(x) = -x + 2\). If we substitute \(x = 3\):
\[ f(3) = -3 + 2 = -1. \]
- The function jumps from \(-6\) to \(-1\) at this point, so there will be an open circle at \((3, -6)\).

From evaluating these points, we can conclude that the following points have open circles:

- \((-3, \frac{1}{2})\)
- \((3, -6)\)

Now, we check the provided options for matching these:

- **(−3,12)**, (3,−1), and (3,−6) – no, as \((-3, 12)\) is not valid and \((3, -1)\) should not be filled in.
- **none of the points** – no, as some points do have open circles.
- **(−3,12), (3,−1), and (3,−6)** – no, as \((-3, 12)\) is incorrect.
- **(−3,12) and (3,−6)** – incorrect for the same reason as above.

Thus, the first statement should have been considered but isn't accurately represented with provided options. Solution implies possible consideration:

- Open circles are at points **(-3, 1/2)** and **(3, -6)**.

Final answer likely would be **“none of the points”** in absence of correct options.