Problem 91 - College Algebra Essentials

Question

$$ T(x)=\left\\{\begin{array}{c} 0.10 x \\ 850.00+0.15(x-8500) \\ 4750.00+0.25(x-34,500) \\ 17,025.00+0.28(x-83,600) \\ \hline \end{array}\right. $$ 10 Find and interpret \(T(20,000)\)

Step-by-Step Solution

Answer

The tax for an income of $20,000 is $2475.

Step 1: Identify the Relevant Piece

The correct piece of the function to use is determined by the income brackets in the piecewise function. For an income, \(x\), of $20,000, we are in the second bracket of the function, \(850.00+0.15(x-8500)\). This is because $20,000 is greater than $8,500 but less than $34,500.

Step 2: Plug-In and Calculate

Plug in the income \(x=20,000\) into the relevant piece: \(T(20,000) = 850.00+0.15(20,000-8500)\). Expanding this you get \(T(20,000)=850.00+0.15 \times 11,500\). Calculating this, you find that \(T(20,000) = 2475\)

Step 3: Interpret the Result

The output of the tax function \(T\) represents the amount of tax to be paid on the income \(x\). Thus, if an individual earns a salary of $20,000, they would be expected to pay $2475 in tax.

Key Concepts

College Algebra

College algebra is a foundational mathematics course that is essential for students in various fields of study. It focuses on enhancing students' understanding of algebraic principles like functions, equations, inequalities, and their applications. A core component of college algebra is working with piecewise functions, which are functions defined by multiple sub-functions, each applying to a certain interval of the main function's domain.

Understanding piecewise functions necessitates recognizing these multiple intervals and the corresponding sub-function that is applicable within each interval. Students learn to analyze complex real-world problems by breaking them down into simpler, more manageable pieces, which is precisely the case with piecewise functions. They epitomize the significance of dissecting a problem based on specific conditions to provide a comprehensive solution.

Function Interpretation

Function interpretation in mathematics revolves around analyzing and making sense of a function's output based on its input values. With a piecewise function like the tax function in the given exercise, function interpretation requires understanding not just the computational output, but the contextual meaning behind it.

When interpreting the function, it’s important to note that different rules apply to different intervals, reflecting the piecewise nature of the tax brackets. For instance, in the exercise provided, interpreting the function value of \(T(20,000)\) means determining how much tax is owed on an income of $20,000. It's not solely about plugging in a number and finding an answer—it's about relating that answer to the real-world context of tax calculation.

Mathematical Calculations

Mathematical calculations in college algebra often involve complex function manipulations, including the evaluation of piecewise functions. Solving these functions accurately requires a systematic approach that includes identifying the correct interval, substituting the input value, and performing the arithmetic operations following the order of operations.

In our exercise example, the student must first determine the correct bracket based on the given income, then insert the value into the correct piece of the function, and carry out the calculations meticulously. Further, understanding how to manipulate and simplify algebraic expressions forms the crux of evaluating piecewise functions. Such skills are indispensable for students not only in academic settings but also in various practical applications where mathematical modeling and calculations are involved.