Problem 86 - College Algebra

Question

The number of lawyers in the United States can be modeled by the function $$ f(x)=\left\\{\begin{array}{ll} 6.5 x+200 & \text { if } 0 \leq x<23 \\ 26.2 x-252 & \text { if } x \geq 23 \end{array}\right. $$ where \(x\) represents the number of years after 1951 and \(f(x)\) represents the number of lawyers, in thousands. In Exercises \(85-88,\) use this function to find and interpret each of the following. $$ f(10) $$

Step-by-Step Solution

Answer

The number of lawyers in the United States 10 years after 1951 was 265 thousand.

Step 1: Identify the Relevant Part of the Function

The function f(x) is defined as two separate functions, a piecewise-defined function, depending on the value of x. Because the value at x=10 falls within the range of 0 ≤ x < 23, the first function that will be used is 6.5x + 200.

Step 2: Calculate the Value of the Function

Substitute x=10 into the function 6.5x + 200. This gives f(10) = 6.5 * 10 + 200, which simplifies to f(10) = 265.

Step 3: Interpret the Result

The value of f(10) = 265 means that 10 years after 1951, there were 265 thousand lawyers in the United States.

Key Concepts

College Algebra

College Algebra is an essential course that provides students with a solid foundation in algebraic principles, including how to work with and understand functions. Piecewise-defined functions are an integral part of College Algebra. They allow us to model situations that can't be represented by a single rule or relationship.

In practice, piecewise-defined functions, such as the one given to model the number of lawyers in the United States, consist of multiple segments or 'pieces', each with their own defining equation and applicable range. To address exercises like finding the value of such a function for a specific input, first, determine which piece of the function to use, and then apply the algebraic skills you've learned to solve for the output.

Modeling with Functions

Modeling with functions is a powerful mathematical tool that helps translate real-world scenarios into mathematical language. Functions are often used to describe the relationship between two quantities, allowing predictions and analyses of various situations. The piecewise-defined function used to model the number of lawyers in the US is a great example of such a tool.

By using different linear equations for different periods (before and after the 23rd year from 1951), the model can adapt to changing rates of increase in the lawyer population. When modeling, it is crucial to understand the context to select the proper function or functions to best describe the situation. The accurate interpretation of function value then provides insights into the modeled phenomenon.

Function Interpretation

Function interpretation involves analyzing the output of a function to provide meaningful information about a real-world situation. In the given problem, interpreting the value of the function at a specific point, such as 10 years after 1951, required us to calculate and then make sense of the result within the context of the problem—number of lawyers in thousands.

To interpret is to take the numerical output and translate it back into the real-world terms it represents. For example, the calculated value of 265 for f(10) does not just indicate a number, but rather a significant historical data point, indicating that there were 265,000 lawyers in the United States ten years after 1951. This insight can inform discussions about social and economic trends related to the legal profession.