A clothing store is having a sale on t-shirts: Buy 4 shirts at full price, then every subsequent shirt is
half off. The total amount a customer spends on x shirts can be modeled by the piecewise function
below:
T(x) = 28x for 0 ≤ x ≤ 4
14x + 56 for x > 4
(a) What is the full price for t-shirts? Explain how you know.
(b) Use the model to find how much a customer would pay for 3 shirts.
(c) Use the model to find how much a customer would pay for 6 shirts.
(d) Sketch a graph of T(x). Find enough points to be able to accurately sketch the graph and show
all your work.
(e) Is the graph continuous? Why or why not
(a) To find the full price for t-shirts, we need to determine the price before any discounts are applied. In this case, the price before any discounts are applied is represented by the function T(x) = 28x for 0 ≤ x ≤ 4. This means that for every shirt purchased within the range of 0 to 4 shirts, the cost per shirt is $28. Therefore, the full price for t-shirts is $28.
(b) To find out how much a customer would pay for 3 shirts, we need to determine the value of T(x) when x = 3. Since 0 ≤ 3 ≤ 4, the function T(x) = 28x applies. Plugging in x = 3 into the function, we get T(3) = 28(3) = 84. Therefore, a customer would pay $84 for 3 shirts.
(c) Similarly, to find out how much a customer would pay for 6 shirts, we need to determine the value of T(x) when x = 6. Since x > 4, we use the second part of the function, T(x) = 14x + 56. Plugging in x = 6 into the function, we get T(6) = 14(6) + 56 = 84 + 56 = 140. Therefore, a customer would pay $140 for 6 shirts.
(d) To sketch the graph of T(x), we need to plot enough points to accurately represent the function. We already know that for 0 ≤ x ≤ 4, the function T(x) = 28x. Therefore, we can plot the points (0, 0), (1, 28), (2, 56), (3, 84), and (4, 112) to represent this part of the graph. For x > 4, we have the function T(x) = 14x + 56. We can plot additional points for this part, such as (5, 126) and (6, 140). Connecting all the plotted points will give us the complete graph of T(x).
(e) The graph of T(x) is continuous. This means that there are no gaps or breaks in the graph. It is continuous because the function is defined for all values of x in its domain, and there are no sudden jumps or discontinuities in the graph. In other words, as x increases or decreases, the graph of T(x) smoothly transitions between different parts of the function.