Problem 44 - Introductory Algebra for College Students

Question

The function \(f(x)=0.76 x+171.4\) models the cholesterol level of an American man as a function of his age, \(x,\) in years. Find and interpret \(f(50)\)

Step-by-Step Solution

Answer

The cholesterol level of an American man of 50 years is expected to be 209.4 units.

Step 1: Substitute x into the given function

To find \(f(50)\), we substitute \(x=50\) into the function \(f(x)=0.76 x+171.4\). So, we get \(f(50)=0.76(50) + 171.4\).

Step 2: Simplify the Expression

Next, multiply 0.76 by 50 and then add 171.4. Calculation results \(f(50) = 38 + 171.4 = 209.4\). This is the numerical value we get when we substitute \(x=50\) into the function.

Step 3: Interpret the Result

The result \(f(50) = 209.4\) when put into the context of the question implies that an American man of 50 years is expected to have a cholesterol level of 209.4 units.

Key Concepts

Linear Functions in Algebra

When we talk about linear functions in algebra, we're referring to a type of function that has a constant rate of change. This is visually represented by a straight line when plotted on a graph. The general form of a linear function is written as \( f(x) = mx + b \), where \( m \) is the slope of the line and \( b \) is the y-intercept, which is the point where the line crosses the y-axis.

In the context of the exercise given, the function \( f(x) = 0.76x + 171.4 \) is a linear function. Here, \( 0.76 \) represents the rate at which the cholesterol level changes with age, and \( 171.4 \) represents the starting value of one's cholesterol level according to the model. A steep slope would indicate a rapid change in cholesterol levels with age, while a smaller slope would indicate a more gradual change.

This concept of linear functions is central in understanding how we can predict cholesterol levels at different ages using simple algebraic operations, which are the foundation of many real-world applications.

Interpreting Function Values

Interpreting function values is about understanding what the output of a function means in a real-world context. The function \( f(x) \) can be seen as a machine where you input a value \( x \), and it gives you back an output \( f(x) \).

In our exercise, to interpret \( f(50) \), we consider the value of \( 50 \) as the age of an American man, and the function output \( 209.4 \) represents his cholesterol level. The interpretation then is that at the age of 50, the model predicts that man will have a cholesterol level of 209.4 units. It's important to note that this value is specific to the model constructed and may not represent all individuals but gives a general idea based on the trend observed in the population.

Understanding how to interpret these values is crucial, as it allows students to apply mathematical models to tangible situations, whether they're analyzing trends in health, economics, or any other field where such models become applicable.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can contain ordinary numbers, variables (like \( x \)), and operations (such as addition and multiplication). The expression \( 0.76x + 171.4 \) from the exercise is an algebraic expression where \( x \) is the variable that represents an individual’s age, and the numbers are the coefficients which scale and adjust the variable to model the cholesterol level.

Algebraic expressions are fundamental in creating equations that model real-world phenomena. They allow us to describe relationship patterns and changes succinctly. The ability to manipulate these expressions, such as substituting in a value for \( x \), is key to finding solutions to problems modelled by algebraic equations.

By understanding algebraic expressions, students can better grasp how altering the value of \( x \) in the function \( f(x) \) affects the outcome, and how such changes reflect real changes in the situation being modeled, like aging and its impact on cholesterol levels.