Problem 108 - Precalculus Essentials

Question

The federal minimum wage was \(\$ 5.15\) per hour from 1997 through \(2006,\) so \(f(x)=5.15\) models the minimum wage, \(f(x)\) in dollars, for the domain \(\\{1997,1998,1999, \ldots, 2006\\}\).

Step-by-Step Solution

Answer

The function \(f(x) = 5.15\) represents the federal minimum wage during the years 1997 to 2006. For any given year within this range, the minimum wage was $5.15.

Step 1: Analyze the Function

Take a look at the function \(f(x) = 5.15\). This is a function that maps every number to a single value, \(5.15\). It's a constant function because the output value doesn't change with different input values.

Step 2: Understand the Domain

The domain of the function is an integral part of understanding the context. In this case, it is the set \(\{1997,1998,1999, \ldots, 2006\}\), which represents the years between 1997 and 2006, inclusive.

Step 3: Interpreting the Function in the Context

Contextually, this function can be interpreted as the federal minimum wage in the U.S. during the years 1997 to 2006. So, for any year \(x\), the value returned by the function represents the minimum wage, which is $5.15 in this scenario.

Key Concepts

Function Analysis

In mathematics, analyzing a function involves understanding the rule or relation that defines it. For a constant function such as \(f(x) = 5.15\), the defining characteristic is its simplicity. Regardless of the input \(x\), the output is always the same: \(5.15\).
The constant value, \(5.15\), tells us that the function does not vary with different inputs. This kind of function is graphically represented by a horizontal line on a graph. No matter what input (or \(x\)-value) you choose, the output (or \(y\)-value) will remain stable.
Understanding constant functions are essential in function analysis because they provide a baseline or reference. They help us to identify when and how other functions vary compared to a constant scenario, like the federal minimum wage in this example.

Domain of a Function

The domain of a function is the set of all possible input values that the function can accept. In our exercise, the domain is the set of years from 1997 to 2006:

• 1997
• 1998
• 1999
• ...
• 2006

Understanding the domain helps to set the limits of the function's application. In this case, it contextualizes the federal minimum wage being applied over a span of years.
It’s crucial to grasp the domain because it gives you boundaries on when and where the function is applicable. In real-life scenarios, like wages over time, understanding the domain allows for precise modeling of the function’s relevance. Without the correct domain, a function might be misconstrued or applied incorrectly.

Contextual Interpretation of Functions

Interpreting a function in context means putting it into real-world scenarios. For the function \(f(x) = 5.15\), it represents a fixed minimum wage over a span of years, specifically from 1997 to 2006. In this situation, each year within the domain sees the same minimum wage.
The value of 5.15 dollars per hour highlights how, regardless of economic conditions, the minimum wage remained constant during that period. This interpretation is vital for understanding how functions can represent real-life situations.
By contextualizing functions, we gain insight into how mathematical concepts translate into everyday events and policies. This insight helps us apply these concepts more effectively and understand their implications in broader contexts like economic policies.