Question

A gas station is offering a deal. If you buy 10 gallons or more of gas, you will be charged $2.25 per gallon instead of the regular $2.50 per gallon. The deal is modeled by the following piecewise function. f(x)={2.5x if x<102.25x if x≥10 If you were to create a graph of this piecewise function, which point would have an open circle? (1 point) Responses (10,22.5) left parenthesis 10 comma 22.5 right parenthesis (1,2.5) left parenthesis 1 comma 2.5 right parenthesis (10,25) left parenthesis 10 comma 25 right parenthesis (0,0)

Answer 1

To analyze the piecewise function given:

\[ f(x) = \begin{cases} 2.5x & \text{if } x < 10 \ 2.25x & \text{if } x \geq 10 \end{cases} \]

We need to determine the behavior of the function around the point \(x = 10\).

1. For \(x < 10\), \(f(x) = 2.5x\). When \(x = 10\), this part does not apply, and the value would be \(2.5(10) = 25\). This means at \(x = 10\) there would be an output of 25 from this equation if it were applicable.

2. For \(x \geq 10\), \(f(x) = 2.25x\). When \(x = 10\): \[ f(10) = 2.25(10) = 22.5 \]

At \(x = 10\), the function transitions from one piece to the other. It would have a value of 25 from the first piece (not included since \(x = 10\) is not less than 10) and a value of 22.5 from the second piece (included). Therefore, there will be an open circle at the point \( (10, 25) \) since that point is not included in the graph of the function.

Thus, the correct answer for the point that would have an open circle is \( (10, 25) \).

Answer 2

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≤31.0 if 3<x≤60.5x if x>6 Does the graph accurately represent the piecewise function? Why or why not? (1 point) Responses Yes. The graph is correct because all three pieces of the function are present. Yes. The graph is correct because all three pieces of the function are present. No. There should be an open circle on the first point of two parts of the function. No. There should be an open circle on the first point of two parts of the function. Yes, because all of the points at the ends of the pieces of the function should have closed circles. Yes, because all of the points at the ends of the pieces of the function should have closed circles. No. The graph should be continuous, with all the pieces connected. No. The graph should be continuous, with all the pieces connected. Skip to navigation

Answer 3

To evaluate whether the graph accurately represents the given piecewise function, we need to inspect the function defined as follows:

\[ f(x) = \begin{cases} 0.5 & \text{if } 1 \leq x \leq 3 \ 1.0 & \text{if } 3 < x \leq 6 \ 0.5x & \text{if } x > 6 \end{cases} \]

Analyzing the Function

1. First Piece (Constant Dosage):

   • For \(1 \leq x \leq 3\), the dosage is 0.5.
   • The endpoints \(x = 1\) and \(x = 3\) are included; thus, there should be closed circles at these points.

2. Second Piece (Constant Dosage):

   • For \(3 < x \leq 6\), the dosage is 1.0.
   • The point \(x = 3\) is not included (since the interval specifies \(3 < x\)), so there should be an open circle at \(x = 3\). The point at \(x = 6\) is included, so it should have a closed circle.

3. Third Piece (Linear Dependency):

   • For \(x > 6\), the dosage is given by \(0.5x\), which is a continuous function for all \(x > 6\).

Conclusion

• There needs to be an open circle at \(x = 3\) because the transition between the first two pieces does not include the value at \(x = 3\) (due to the \(3 < x\) condition).
• Also, there should be a closed circle at \(x = 6\).

Answer Summary

Given this analysis, the correct response would be:

No. There should be an open circle on the first point of two parts of the function.

This accurately reflects the requirement of the piecewise function and how it should be graphed.