To determine the error in Jayden's graph of the piecewise function \( f(x) = \begin{cases} -x + 1 & \text{if } x < 0 \ 0 & \text{if } 0 \leq x \end{cases} \), we need to analyze the function based on its pieces.
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For \( x < 0 \), the equation is \( f(x) = -x + 1 \). The graph of this line can be evaluated at specific points:
- At \( x = -1 \), \( f(-1) = -(-1) + 1 = 2 \).
- At \( x = 0 \), no value is assigned because this part of the function is only valid for \( x < 0 \).
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For \( x \geq 0 \), the function is constant: \( f(x) = 0 \).
Now let's see where the points fall:
- At \( x = 0 \), since that belongs to the second piece of the function \( (0 \leq x) \), we have \( f(0) = 0 \). This point should be a closed dot at (0, 0).
- There is no defined value for \( f(x) \) at \( x = 0 \) coming from the first piece, so the point (0, 1) should not be part of the graph (meaning an open dot should not be there).
Now, let's evaluate the options for errors:
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Option #1: The graph should have a closed dot at (0, 1).
- This is incorrect. There should not be a closed dot at (0, 1); instead, there should be a dot at (0, 0).
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Option #2: The graph should have an open dot at (0, −2).
- Incorrect. No point at (0, −2) exists.
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Option #3: The point at (0, −2) should be an arrow to the left.
- Incorrect. There’s no such point in the defined piecewise function.
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Option #4: The point at (−2, 3) should be an arrow to the left.
- Incorrect. There is no point defined at \( (−2, 3) \) in the function, so this is also a mistake.
Since none of the options correctly describe the error based on the piecewise function’s requirements. The correct statement would be that there should be a closed dot at (0, 0) and an open dot at (0, 1) should not be present. However, since none of the options match that, if you meant (0, 1) as the closed point location, it would make Option #1 an interpretation error rather than grappling the piecewise definition accurately.
If you are considering which would be the best description from the provided options, the function should effectively reflect the proper representation and show that (0, 0) is indeed correct while noting the absence of a closed dot at (0, 1) is crucial.
None of the provided options therefore completely capture the desired error, but if basing a decision on typical conventions where Option #1 implies a misunderstanding in closure vs. openness of (0, 1) as declarative, it can be leveraged for adjustment to show the closure actually should be made at (0, 0).
Thus, I recommend advising: Option #1: Should be corrected as the graph should have a closed dot at (0, 0) and none other is needed. - adjust as required.