Problem 90 - A Graphical Approach to Precalculus with Limits

Question

Function \(f\) gives the median 2015 weekly income (in dollars) by educational attainment for people 25 years old and over. This function is defined by \(f(N)=493, f(H)=678, f(B)=1137\) and \(f(M)=1341,\) where \(N\) denotes no high school diploma, \(H\) a high school diploma, \(B\) a bachelor's degree, and \(M\) a master's degree. (Source: U.S. Bureau of Labor Statistics.) (a) Write \(f\) as a set of ordered pairs. (b) Give the domain and range of \(f\). (c) Discuss the relationship between education and income.

Step-by-Step Solution

Answer

(a) \((N, 493), (H, 678), (B, 1137), (M, 1341)\); (b) Domain: \(\{N, H, B, M\}\), Range: \(\{493, 678, 1137, 1341\}\); (c) Higher education generally leads to higher income.

Step 1: Define the Function as Ordered Pairs

To express the function \(f\) as a set of ordered pairs, pair each educational level with its corresponding income value. The function \(f\) gives us:- \(N\) maps to 493, - \(H\) maps to 678, - \(B\) maps to 1137, - \(M\) maps to 1341. Thus, the set of ordered pairs is \(\{(N, 493), (H, 678), (B, 1137), (M, 1341)\}\).

Step 2: Determine the Domain of the Function

The domain of a function is the set of all possible inputs. In this function \(f\), the inputs are the levels of educational attainment. Therefore, the domain of \(f\) is \(\{N, H, B, M\}\).

Step 3: Determine the Range of the Function

The range of a function is the set of all possible output values. For this function, the outputs are the weekly incomes. Therefore, the range of \(f\) is \(\{493, 678, 1137, 1341\}\).

Step 4: Analyze the Relationship between Education and Income

By examining the function's ordered pairs, we see that as educational attainment increases, the median 2015 weekly income also increases. This suggests a positive correlation between higher education levels and higher income, indicating that higher education typically leads to better-paying jobs.

Key Concepts

Ordered Pairs

In precalculus, we often use ordered pairs to represent functions and relationships between two sets. An ordered pair consists of two elements: a first element known as the domain, and a second element called the range. In our function \( f \), this setup allows us to see a direct relationship between educational attainment, represented by 'N', 'H', 'B', and 'M', and their corresponding incomes in dollars.Ordered pairs can be thought of like coordinates on a graph; the first element is like the x-coordinate, and the second is the y-coordinate:

• \((N, 493)\)
• \((H, 678)\)
• \((B, 1137)\)
• \((M, 1341)\)

Here, each educational level \((N, H, B, M)\) is paired with its respective median income, providing a clear picture of how education level and income are related.

Domain and Range

The concepts of domain and range are fundamental in understanding functions. The domain includes all possible inputs for the function, while the range consists of all potential outputs.For the function \( f \) regarding educational levels and incomes:

• The domain is \( \{N, H, B, M\} \). This means that our inputs are the different levels of educational attainment: No high school diploma, High school diploma, Bachelor's degree, and Master's degree.
• The range is \( \{493, 678, 1137, 1341\} \). These outputs represent the median weekly incomes corresponding to the education levels, showing the monetary earnings associated with each educational qualification.

By defining the domain and range, we can clearly understand the scope and limitations of our function \( f \). We know exactly what inputs are allowable and what kind of outputs we can expect.

Correlation between Education and Income

The exercise highlights an important real-world application of mathematical functions: analyzing the correlation between education and income.Educational attainment often plays a significant role in determining income levels. In our function \( f \), there's a clear positive correlation between higher educational attainment and increased income:

• People without a high school diploma \((N)\) have the lowest income \(493\) dollars per week.
• Income increases with a high school diploma \((H)\) at \(678\) dollars per week.
• A Bachelor's degree \((B)\) typically brings in \(1137\) dollars per week.
• Lastly, with a Master's degree \((M)\), one can expect to earn about \(1341\) dollars per week.

This trend suggests that investing in education can lead to better job opportunities and higher earnings. Analyzing this data shows the importance of education in achieving financial success, which can be a motivating factor for pursuing higher degrees.