Problem 29 - Algebra Form and Function

Question

Table 4.12 shows monthly life insurance rates, in dollars, for men and women. Let \(m=f(a)\) be the rate for men at age \(a\), and \(w=g(a)\) be the rate for women at age \(a\). (a) Find \(f(65)\). (b) Find \(g(50)\). (c) Solve and interpret \(f(a)=102\). (d) Solve and interpret \(g(a)=57\). (e) For what values of \(a\) is \(f(a)=g(a) ?\) (f) For what values of \(a\) is \(g(a)

Step-by-Step Solution

Answer

Answer: g(a) < f(a) for a = 50, 55, 60, 65, and 70.

Step 1: Locate Age in the Table

Find the age 65 in the first row of the table.

Step 2: Find the Corresponding Rate for Men

Locate the rate for men at age 65 in the row below the first row (the \(f(x)\) row). The value of \(f(65)\) is 218. #b) Finding g(50)#

Step 1: Locate Age in the Table

Find the age 50 in the first row of the table.

Step 2: Find the Corresponding Rate for Women

Locate the rate for women at age 50 in the row below the row for men (the \(g(x)\) row). The value of \(g(50)\) is 39. #c) Solve and Interpret f(a) = 102#

Step 1: Locate the Value 102 in the f(x) Row

Find the value 102 in the \(f(x)\) row (the row for men).

Step 2: Find the Corresponding Age

Locate the age in the first row that corresponds to the value 102 in the \(f(x)\) row. The age is 60. Interpretation: The life insurance rate for men is $102 when the age is 60. #d) Solve and Interpret g(a) = 57#

Step 1: Locate the Value 57 in the g(x) Row

Find the value 57 in the \(g(x)\) row (the row for women).

Step 2: Find the Corresponding Age

Locate the age in the first row that corresponds to the value 57 in the \(g(x)\) row. The age is 55. Interpretation: The life insurance rate for women is $57 when the age is 55. #e) For What Values of a is f(a) = g(a)?#

Step 1: Compare Rows f(x) and g(x)

Look at the rows for men and women and find the values where both rows (rates for men and women) are equal.

Step 2: List the Ages Where Rates are Equal

f(a) = g(a) for a = 35, 40, and 45. #f) For What Values of a is g(a) < f(a)?#

Step 1: Compare Rows f(x) and g(x)

Look at the rows for men and women and find the values where the row for women (g(x)) is less than the row for men (f(x)).

Step 2: List the Ages Where Rates for Women are Less Than Rates for Men

g(a) < f(a) for a = 50, 55, 60, 65, and 70.

Key Concepts

Functions

In this exercise, the concept of a function is at the heart of understanding how life insurance rates are structured based on age. Functions allow us to express a relationship between two quantities: in this case, age and insurance rate.

We denote the insurance rate for men as \( f(a) \) and for women as \( g(a) \), where \( a \) represents age. The function \( f(a) \) provides the insurance rate charged to men of age \( a \), while \( g(a) \) does the same for women.

Functions like \( f(a) \) and \( g(a) \) are useful because they provide a clear, predictable way of determining insurance rates based on age. Understanding the notation and how to work with these functions is crucial for interpreting data about insurance rates and making comparisons.

Age Comparison

In the context of this exercise, age comparison involves analyzing and comparing the insurance rates for different ages, and understanding at what age males and females experience equivalent or differing rates.

When examining the table, you can see that certain ages have the same insurance rates for both men and women, specifically at ages 35, 40, and 45. This equal rate is expressed as \( f(a) = g(a) \).

Additionally, the concept of age comparison helps to determine at which ages rates for women are less than those for men. For instance, from age 50 onward, women's rates noted as \( g(a) < f(a) \) are consistently lower than men's rates, highlighting a trend that could be tied to various factors such as risk assessment made by insurers.

Age comparison is essential for evaluating fairness and determining how rates change as individuals age.

Data Interpretation

Data interpretation is essential for translating the tabulated information on life insurance rates into real-world insights.

This process involves reading and extracting meaning from the data presented in the table. For instance, locating values such as \( f(65) \) informs you that a 65-year-old man's life insurance rate is \(218. Similarly, understanding \( g(50) \) allows you to know that a 50-year-old woman pays \)39 for her life insurance.

Data interpretation also covers understanding what it means when functions meet specific values (e.g., \( f(a) = 102 \) or \( g(a) = 57 \)) and realizing its implication regarding age and rate equivalence as seen in \( f(a) = g(a) \) scenarios.

Essentially, this skill allows one to make informed decisions and analyses based on available quantitative data, ensuring interpretations are accurate and useful.