Question
Children’s cough syrup has dosage guidelines based on a child’s age. In the following piecewise function, x represents the child’s age in years and f(x) represents the dosage in fluid ounces.
F(x) = { 0.5 if 1 < x < 3
1.0 if 3 < x < 6
0.5x if x > 6
Does the graph accurately represent the piecewise function? Why or why not?
A. Yes, because all of the points at the end of the pieces of the function should have closed circles
B. No. There should be an open circle on the first point of two parts of the function
C. Yes. The graph is correct because all three pieces of the function are present
No. The graph should be continuous, with all the pieces connected
F(x) = { 0.5 if 1 < x < 3
1.0 if 3 < x < 6
0.5x if x > 6
Does the graph accurately represent the piecewise function? Why or why not?
A. Yes, because all of the points at the end of the pieces of the function should have closed circles
B. No. There should be an open circle on the first point of two parts of the function
C. Yes. The graph is correct because all three pieces of the function are present
No. The graph should be continuous, with all the pieces connected
Answers
GPT-4o mini
To determine if the graph accurately represents the piecewise function given, we need to carefully analyze the specified conditions of the function and how the graph should represent them.
The piecewise function is defined as follows:
1. \( F(x) = 0.5 \) if \( 1 < x < 3 \): This means the dosage is \( 0.5 \) ounces for ages greater than 1 year and less than 3 years. Both endpoints (1 and 3) are not included, so there should be open circles at \( x = 1 \) and \( x = 3 \).
2. \( F(x) = 1.0 \) if \( 3 < x < 6 \): Here, the dosage is \( 1.0 \) ounces for ages greater than 3 years and less than 6 years. Similarly, there should be open circles at \( x = 3 \) and \( x = 6 \).
3. \( F(x) = 0.5x \) if \( x > 6 \): This piece is defined for ages greater than 6 years. Since there is no upper limit, this part can be graphed continuously.
Now let's analyze the options:
A. **Yes, because all of the points at the end of the pieces of the function should have closed circles**: This statement is false because at the endpoints (1, 3, and 6), the function does not include those values, hence, they should be open circles.
B. **No. There should be an open circle on the first point of two parts of the function**: This is true because both 3 (where it transitions from 0.5 to 1.0) and 6 (where it transitions from 1.0 to 0.5x) should have open circles.
C. **Yes. The graph is correct because all three pieces of the function are present**: This statement is incorrect if the graph does not include the open circles as specified.
D. **No. The graph should be continuous, with all the pieces connected**: This is false; the graph is not continuous at the points where the function switches from one piece to another (1, 3, and 6) because it has specified open intervals.
Based on this analysis, the correct answer is:
**B. No. There should be an open circle on the first point of two parts of the function.**
This is the most accurate representation of what the piecewise function entails, particularly at the transition points.
The piecewise function is defined as follows:
1. \( F(x) = 0.5 \) if \( 1 < x < 3 \): This means the dosage is \( 0.5 \) ounces for ages greater than 1 year and less than 3 years. Both endpoints (1 and 3) are not included, so there should be open circles at \( x = 1 \) and \( x = 3 \).
2. \( F(x) = 1.0 \) if \( 3 < x < 6 \): Here, the dosage is \( 1.0 \) ounces for ages greater than 3 years and less than 6 years. Similarly, there should be open circles at \( x = 3 \) and \( x = 6 \).
3. \( F(x) = 0.5x \) if \( x > 6 \): This piece is defined for ages greater than 6 years. Since there is no upper limit, this part can be graphed continuously.
Now let's analyze the options:
A. **Yes, because all of the points at the end of the pieces of the function should have closed circles**: This statement is false because at the endpoints (1, 3, and 6), the function does not include those values, hence, they should be open circles.
B. **No. There should be an open circle on the first point of two parts of the function**: This is true because both 3 (where it transitions from 0.5 to 1.0) and 6 (where it transitions from 1.0 to 0.5x) should have open circles.
C. **Yes. The graph is correct because all three pieces of the function are present**: This statement is incorrect if the graph does not include the open circles as specified.
D. **No. The graph should be continuous, with all the pieces connected**: This is false; the graph is not continuous at the points where the function switches from one piece to another (1, 3, and 6) because it has specified open intervals.
Based on this analysis, the correct answer is:
**B. No. There should be an open circle on the first point of two parts of the function.**
This is the most accurate representation of what the piecewise function entails, particularly at the transition points.