Solve each problem. See Example 4. Suppose that the cost of mailing a letter weighing x ounces, where x>0, is ƒ(x)=55-15[[1-x]]cents. What is the cost of mailing a 2.6-ounce letter?

Verified step by step guidance

1. First, understand the function given: \(f(x) = 55 - 15 \times \lceil 1 - x \rceil\), where \(\lceil \cdot \rceil\) denotes the ceiling function, which rounds a number up to the nearest integer.
2. Identify the value of \(x\) for the problem: here, \(x = 2.6\) ounces, which is the weight of the letter.
3. Calculate the expression inside the ceiling function: compute \$1 - x\(, so \)1 - 2.6$.
4. Apply the ceiling function to the result from the previous step: find \(\lceil 1 - 2.6 \rceil\).
5. Substitute the value of the ceiling function back into the original function \(f(x)\) and simplify to find the cost of mailing the 2.6-ounce letter.

Key Concepts

Piecewise and Step Functions

Step functions assign constant values over intervals and often involve the greatest integer (floor) function. Understanding how these functions behave, especially with the floor function, is essential to evaluate expressions like ƒ(x) = 55 - 15⌊1 - x⌋ correctly.

Greatest Integer (Floor) Function

The floor function, denoted ⌊x⌋, returns the greatest integer less than or equal to x. It is crucial to correctly compute ⌊1 - x⌋ for a given x, as it affects the value of the cost function in the problem.

Function Evaluation

Evaluating a function means substituting the given input value into the function's formula and simplifying. Here, substituting x = 2.6 into ƒ(x) requires careful calculation of the floor term and arithmetic to find the mailing cost.