Problem 77 - Algebra and Trigonometry

Question

Internet Purchases An Internet bookstore charges \(15 shipping for orders under \)100 but provides free shipping for orders of \(100 or more. The cost C of an order is a function of the total price x of the books purchased, given by \)C(x)=\left\\{\begin{array}{ll}{x+15} & {\text { if } x<100} \\ {x} & {\text { if } x \geq 100}\end{array}\right.\( (a) Find \)C(75), C(90), C(100),\( and \)C(105)$ (b) What do your answers in part (a) represent?

Step-by-Step Solution

Answer

The total costs are: 90, 105, 100, and 105, for purchases of 75, 90, 100, and 105, respectively.

Step 1: Determine C(75)

Since 75 is less than 100, use the formula \( C(x) = x + 15 \). Substitute \( x = 75 \) into the equation: \( C(75) = 75 + 15 = 90 \).

Step 2: Calculate C(90)

Since 90 is also less than 100, apply the same formula: \( C(x) = x + 15 \). Substitute \( x = 90 \): \( C(90) = 90 + 15 = 105 \).

Step 3: Evaluate C(100)

For \( x = 100 \), which is equal to 100, use the formula \( C(x) = x \). Thus, \( C(100) = 100 \).

Step 4: Compute C(105)

Since 105 is greater than 100, again use the formula \( C(x) = x \). Therefore, \( C(105) = 105 \).

Step 5: Interpret the Results

The values calculated \( C(75) = 90 \), \( C(90) = 105 \), \( C(100) = 100 \), and \( C(105) = 105 \) represent the total cost of orders at different book purchase values including applicable shipping fees. Orders under 100 incur a 15 shipping charge, while those 100 or more are free.

Key Concepts

Function Notation

Function notation is a convenient way to represent mathematical functions and is commonly used in algebra and calculus. In essence, it tells you which function you’re dealing with and what input it's being applied to. The notation typically involves a function name followed by parentheses that enclose the input variable—like this: \( f(x) \). Here, \( f \) is the name of the function and \( x \) is the input or 'independent variable'.

• Function Name: Denotes the specific process or rule describing the relationship between inputs and outputs.
• Input Variable: The value that you substitute into the function's rule to get an output.
• Output Value: The result you get after applying the function rule to the input.

Using function notation, we can easily express relationships and dependencies, keeping our work clear and organized. In the context of piecewise functions, function notation makes it simple to define different rules for different ranges of the input variable.

Evaluating Functions

Evaluating a function means calculating the output value of a function for a given input value. This is done by substituting the input value into the function notation and simplifying according to the function's rules. In the provided exercise with the bookstore's pricing rule, evaluating functions helps determine the total cost for specific amounts of purchase.

Steps to Evaluate a Function

To evaluate a function, follow these steps:

• Identify the proper expression or rule for the given input range. This is crucial for piecewise functions where different rules apply to different input intervals.
• Substitute the input value into the appropriate expression.
• Simplify the expression to get the output.

This approach helps answer the exercise’s question part (a), such as finding \( C(75) \), \( C(90) \), \( C(100) \), and \( C(105) \), making it clear how function evaluation is applied in real-world scenarios like calculating costs.

Conditional Equations

Conditional equations are a type of mathematical statement that includes multiple conditions or sets of rules, often used to express functions with different behavior based on the input value. In piecewise functions, conditional equations determine which mathematical rule applies depending on the condition met by the input value.

How Conditional Equations Work

In the internet bookstore example, the conditional equation defines costs based on whether the total order is below \( 100 \) or not:

• If \( x < 100 \): The equation adds shipping, calculated as \( C(x) = x + 15 \).
• If \( x \geq 100 \): No extra shipping cost is added, so \( C(x) = x \).

This way of structuring equations allows for the modeling of real-world conditions, creating flexible solutions for different situations. Understanding these equations is crucial for interpreting what the output represents, as shown in part (b) of the exercise, describing how charges apply based on the purchase amount.