Problem 92 - A Graphical Approach to College Algebra

Question

Income and Education Function \(f\) gives the aver- age 2010 individual income (in dollars) by educational attainment for people 25 years old and over. This function is defined by \(f(N)=21,484, f(H)=31,286\) \(f(B)=57,181,\) and \(f(M)=70,181,\) 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) \((N, 21484), (H, 31286), (B, 57181), (M, 70181)\); (b) Domain: \(\{N, H, B, M\}\), Range: \(\{21484, 31286, 57181, 70181\}\); (c) Higher education correlates with higher income.

Step 1: Identify Ordered Pairs

The function \(f\) is given in terms of education level as inputs and corresponding income as outputs. To write \(f\) as a set of ordered pairs, pair each education level symbol with its associated income value. The ordered pairs are: \((N, 21484), (H, 31286), (B, 57181), (M, 70181)\).

Step 2: Determine the Domain

The domain of a function is the set of all possible input values. For \(f\), the input values are the education level symbols: \(\{N, H, B, M\}\).

Step 3: Determine the Range

The range of a function is the set of all possible output values. For \(f\), these are the income amounts associated with each education level: \(\{21484, 31286, 57181, 70181\}\).

Step 4: Analyze the Relationship between Education and Income

Observe how the income values increase with higher educational attainments. This shows a positive relationship where individuals with higher levels of education tend to have higher incomes, according to the data provided.

Key Concepts

Domain and Range

When we talk about the "Domain" of a function, we're referring to all the possible input values that the function can accept. In the context of our exercise, the inputs are different levels of educational attainment represented by symbols:

• \( N \) - No high school diploma
• \( H \) - High school diploma
• \( B \) - Bachelor's degree
• \( M \) - Master's degree

Thus, the domain of the function \( f \) can be expressed as the set \( \{ N, H, B, M \} \).

The "Range," on the other hand, refers to all the possible output values of the function. For our function \( f \), the outputs are the average income levels corresponding to each education level:

• \( 21,484 \) - Income for no high school diploma
• \( 31,286 \) - Income for high school diploma
• \( 57,181 \) - Income for bachelor's degree
• \( 70,181 \) - Income for master's degree

Consequently, the range of the function \( f \) is \( \{ 21,484, 31,286, 57,181, 70,181 \} \). Understanding the domain and range is crucial because it tells us the scope of the function.

Ordered Pairs

Ordered pairs represent a fundamental concept in mathematics, especially when we delve into functions and coordinate systems. Let's break down what ordered pairs mean in this context.

In our exercise, the educational levels are paired with their corresponding income values. This pairing creates what we call "ordered pairs." For the function \( f \), which links education to income, the ordered pairs illustrate specific connections between each education level and its average income from 2010. The ordered pairs for this exercise are:

• \( (N, 21,484) \)
• \( (H, 31,286) \)
• \( (B, 57,181) \)
• \( (M, 70,181) \)

Each ordered pair consists of two elements:

• The first element is the independent variable, which in this case is the education level.
• The second element is the dependent variable, representing the income associated with that education level.

Ordered pairs are not just theoretical constructs; they help visualize and understand mathematical relationships in real-world contexts.

Education and Income Relationship

There's a clear relationship between education and income depicted by the function \( f \) in our exercise. This function is an example of how education levels can be directly linked to income levels. Let's explore this relationship.

The data presented shows average income values rising as education levels increase. This indicates a positive correlation between the two variables:

• No high school diploma (\( N \)) corresponds to the lowest average income.
• Obtaining a high school diploma (\( H \)) results in a higher average income than having none.
• A bachelor's degree (\( B \)) leads to an even higher average income.
• Finally, a master's degree (\( M \)) attains the highest average income in the data provided.

This trend supports the general notion that higher education can potentially lead to better-paying jobs. By investing in higher education, individuals might increase their earning potential significantly.

Such insights are valuable for policy makers, educators, and individuals planning their educational paths. They highlight the importance of access to education and its role in improving earning capacity and overall quality of life.