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 for the first ounce?

Verified step by step guidance

1. First, understand the function given: \(f(x) = 55 - 15 \times \lceil 1 - x \rceil\), where \(x > 0\) and \(\lceil \cdot \rceil\) denotes the ceiling function, which rounds a number up to the nearest integer.
2. To find the cost for the first ounce, substitute \(x = 1\) into the function: \(f(1) = 55 - 15 \times \lceil 1 - 1 \rceil\).
3. Calculate the expression inside the ceiling function: \$1 - 1 = 0\(, so \)\lceil 0 \rceil = 0$ because the ceiling of zero is zero.
4. Now, multiply \$15\( by the ceiling value: \)15 \times 0 = 0$.
5. Finally, subtract this product from 55 to find the cost for the first ounce: \$55 - 0$.

Key Concepts

Greatest Integer Function (Floor Function)

The greatest integer function, denoted by [[x]], returns the largest integer less than or equal to x. It is used to round down values to the nearest whole number, which is essential in piecewise or stepwise functions like the cost function given.

Function Evaluation

Function evaluation involves substituting a specific input value into the function's formula to find the corresponding output. Here, evaluating ƒ(x) at x = 1 (the first ounce) requires careful substitution and simplification.

Domain and Constraints

Understanding the domain (x > 0) ensures that the input values are valid for the problem context. Recognizing constraints helps avoid errors, such as evaluating the function at invalid points or misinterpreting the function's behavior.