Problem 16 - Algebra 2 and Trigonometry

Question

The sales tax \(t\) on a purchase is a function of the amount \(a\) of the purchase. The sales tax rate in the city of East chester is 8\(\% .\) a. Write a rule in function notation that can be used to determine the sales tax on a pur- chase in East chester. b. What is a reasonable domain for this function? c. Find the sales tax when the purchase is \(\$ 5.00 .\) d. Find the sales tax when the purchase is \(\$ 16.50 .\)

Step-by-Step Solution

Answer

Function: \( t(a) = 0.08a \), Domain: \( a \geq 0 \), Tax for \$5.00: \$0.40, Tax for \$16.50: \$1.32.

Step 1: Writing the Function

The sales tax can be represented by a function \( t(a) \), where \( a \) is the purchase amount. Since the tax rate is 8\%, the function will be \( t(a) = 0.08a \). This function will compute the tax by multiplying the purchase amount by the tax rate.

Step 2: Identifying the Domain

The domain of the function \( t(a) \) is all non-negative real numbers because a purchase amount cannot be negative. Therefore, the domain is \( a \geq 0 \).

Step 3: Calculating Tax for a $5.00 Purchase

Substitute \( a = 5.00 \) into the function \( t(a) = 0.08a \). This gives \( t(5.00) = 0.08 \times 5.00 = 0.40 \). Thus, the sales tax for a \( \\( 5.00 \) purchase is \( \\) 0.40 \).

Step 4: Calculating Tax for a $16.50 Purchase

Substitute \( a = 16.50 \) into the function \( t(a) = 0.08a \). This gives \( t(16.50) = 0.08 \times 16.50 = 1.32 \). Thus, the sales tax for a \( \\( 16.50 \) purchase is \( \\) 1.32 \).

Key Concepts

Function Notation

Function notation is a way to express the relationship between two variables, usually one representing input and the other representing output. In this exercise, we represent the sales tax as a function of the purchase amount. The function is written as \( t(a) \), where:

• \( t \) is the tax function
• \( a \) is the amount of the purchase

The notation \( t(a) = 0.08a \) means that to find the tax, you multiply the purchase amount \( a \) by the tax rate, which is 8% or 0.08 in decimal form. This makes it clear that every dollar of purchase contributes 8 cents to the tax. The beauty of function notation lies in its simplicity and clarity, letting us quickly identify how one quantity affects another.

Domain and Range in Mathematics

In mathematics, the domain and range of a function describe the set of possible input and output values, respectively. They are essential for understanding the limitations and behavior of a function.

• Domain: For the function \( t(a) = 0.08a \), the domain is all possible values that \( a \), the purchase amount, can take. Since you can't have a negative purchase, the domain is all non-negative real numbers, or \( a \geq 0 \).
• Range: The range is determined by the possible output values of \( t(a) \). Since the smallest tax is when \( a = 0 \), giving \( t(0) = 0 \), and there's no theoretical maximum, the range is all non-negative real numbers as well, or \( t(a) \geq 0 \).

Understanding the domain and range helps prevent errors when evaluating the function with inputs that don't make sense in real-world scenarios.

Real Numbers

Real numbers are a broad category of numbers used to denote any value along the continuous number line, including fractions and irrational numbers.
This concept is vital in mathematics because it allows us to work with not just whole numbers, but also all decimals and fractions between them.
When we talk about the domain of the sales tax function, which is restricted to real numbers, it means that any real, non-negative number can be a potential input for purchase amount \( a \).
Real numbers ensure that our calculations for sales tax can accommodate any realistic purchase value, from a tiny fraction of a dollar to large sums for expensive purchases, making the function flexible and applicable in varied scenarios.