Problem 85 - 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. $$ f(0) $$

Step-by-Step Solution

Answer

The number of lawyers in the United States in the year 1951 (0 years after 1951) was approximately 200,000.

Step 1: Rule Selection

Depending on the given input, choose the appropriate function to use. Here \(x = 0\), so the function that is valid for this range is \(6.5x + 200\).

Step 2: Function Evaluation

Substitute \(x = 0\) into the chosen function to get \(f(0) = 6.5 * 0 + 200\).

Step 3: Simplify and Solve

Solve the expression to get \(f(0)\). In this case, \( f(0)= 200 \).

Key Concepts

College Algebra

College Algebra is a foundational course that emphasizes the development of skills and understanding of algebraic concepts, which are fundamental to further study in mathematics and its applications in various fields. One of the key elements of college algebra is working with piecewise functions, like the one given in our example.

In our example, the function for the number of lawyers in the United States is defined by two different expressions, depending on the value of the input, which is the number of years after 1951. When working with these functions, it is vital to determine which expression to use based on the value of the input, as was done in the step-by-step solution. This exercise requires students to interpret both the function as a whole and the various components of the function, enhancing their algebraic reasoning and critical thinking skills.

Function Modeling

Function modeling entails creating a mathematical representation of a real-world situation. The function provided in the exercise serves as a model to estimate the number of lawyers in the United States over time. Real-world phenomena are often complex and cannot be captured by a single mathematical expression across all situations; hence, piecewise functions are used to model different behaviors or conditions.

In our exercise, the piecewise function shows a change of rate at a specific point in time, 23 years after 1951. Understanding and creating such models is essential because it allows students to translate a verbal description of a data trend into a mathematical form, enable predictions, and make informed decisions based on those predictions. Effective modeling requires a blend of conceptual understanding and practical application of algebraic skills, both of which are reinforced by practicing with functions like the one in our exercise.

Interpreting Functions

Interpreting functions involves more than just plugging in values and solving; it is about understanding what the function is conveying about a real-world context. When interpreting the piecewise function for the number of lawyers, recognizing the point at which the model changes and what that change means in the real-world context is key.

For example, in calculating and interpreting the value of \(f(0)\), students not only apply algebraic operations but also infer that at the starting year, 1951, there were 200 thousand lawyers in the United States. This ability to move from abstract mathematical processes to concrete, meaningful conclusions is an invaluable part of learning algebra. It involves synthesizing numerical results and the context of the function, a practice that enhances mathematical literacy and comprehension across various disciplines and applications.