A store is having a sale on boxes of cereal. In the following piecewise function, the independent variable represents the number of boxes purchased. If fewer than 5 boxes are purchased, the cost per box is $3.50. If 5 or more boxes are purchased, the cost per box is only $2.25

Question

f(x) ={3.50 if <=x<5}
{2.25if x>=5}
Does graph represent the situation why or why not

Answer 1

The piecewise function \( f(x) \) you've described represents the cost per box of cereal based on the number of boxes \( x \) purchased. Let's examine it in detail and then talk about its graphical representation.

Your function can be expressed as:

\[ f(x) = \begin{cases} 3.50 & \text{if } x < 5 \ 2.25 & \text{if } x \geq 5 \end{cases} \]

Analysis of the Function:

1. For \( x < 5 \): When the number of boxes purchased is fewer than 5, the cost per box is $3.50. This means that if someone buys 0, 1, 2, 3, or 4 boxes, the cost per box remains constant at $3.50.

2. For \( x \geq 5 \): When the number of boxes purchased is 5 or more, the cost per box drops to $2.25. Therefore, if someone buys 5, 6, 7 or more boxes, the cost per box remains constant at $2.25.

Graph of the Function:

• The graph of this piecewise function would consist of two horizontal lines:
  • A horizontal line at \( y = 3.50 \) for \( x \) values from 0 to just below 5. This line would not include the point at \( x = 5\) since that point falls into the next piece of the function.
  • A horizontal line at \( y = 2.25 \) starting from \( x = 5 \) and going to the right (i.e., for \( x = 5\) and greater). The point at \( x = 5 \) would be included in this line.

Why the Graph Accurately Represents the Situation:

• Clarity in Pricing Structure: The graph visually demonstrates how the price per box changes based on the quantity purchased, clearly indicating which pricing applies for different ranges of \( x \).
• Continuity and Values: The jump from $3.50 to $2.25 at \( x = 5 \) is clearly marked, clearly communicating the pricing strategy of the store.
• Appropriate Representation of Variables: The independent variable (boxes purchased) and the dependent variable (cost per box) are appropriately represented, with \( x \) on the horizontal axis and \( f(x) \) on the vertical axis.

Conclusion:

Yes, the graph represents the situation accurately because it clearly depicts the varying cost per box of cereal based on the number of boxes purchased. The piecewise function captures the rules governing the pricing strategy, and the graph visually illustrates these rules effectively.