Problem 74 - Algebra and Trigonometry

Question

Examples of Functions At the beginning of this section we discussed three examples of everyday, ordinary functions: Height is a function of age, temperature is a function of date, and postage cost is a function of weight. Give three other examples of functions from everyday life.

Step-by-Step Solution

Answer

Examples: Distance as a function of time, salary as a function of hours worked, and electricity bill as a function of usage.

Step 1: Understand the Definition of a Function

Firstly, recall that a function is a relationship between two sets where each input from the first set (domain) is related to exactly one output from the second set (range). With this understanding, we can consider examples where one variable depends on another.

Step 2: Analyze Everyday Variables

Think about variables in daily life where a change in one directly affects the other. This could relate to measurements, costs, or time-dependent events.

Step 3: Identify Example 1 - Distance as a Function of Time

Consider "distance traveled" as a function of "time spent traveling." For a vehicle moving at a constant speed, distance changes as time progresses. The equation for this can be expressed as \( \text{Distance} = \text{Speed} \times \text{Time} \).

Step 4: Identify Example 2 - Salary as a Function of Hours Worked

In most jobs, the salary earned is directly related to the number of hours worked, assuming an hourly wage. The more hours worked, the higher the salary, which can be expressed as \( \text{Salary} = \text{Hourly Rate} \times \text{Hours Worked} \).

Step 5: Identify Example 3 - Electricity Bill as a Function of Electricity Used

Your electricity bill amount is a function of the amount of electricity consumed during a billing period. More electricity usage results in a higher bill amount. This can be represented by \( \text{Bill Amount} = \text{Rate per kWh} \times \text{kWh Used} \).

Key Concepts

Relationships between Variables

In mathematics, a function represents a special type of relationship between two sets of items. This relationship involves a dependency, where each element from one set (called the domain) is linked to a single element in another set (the range). Understanding this concept can clarify how different variables are related in various scenarios.
For instance, consider the example of distance being a function of time. When you travel in a car at a constant speed, the distance you cover is directly linked to how much time you spend traveling. This creates a predictable relationship captured in the formula: \( \text{Distance} = \text{Speed} \times \text{Time} \).
Recognizing these relationships helps in visualizing how changing one variable affects another. Here are a few key aspects of these relationships:

• One-to-one relationship: Each input has one unique output.
• A consistent pattern or rule defines the relationship.

Understanding functions as relationships between variables lays the foundation for exploring more complex mathematical concepts.

Real-life Examples of Functions

Functions are not just theoretical constructs confined to classrooms. They frequently appear in real life, helping us describe a variety of scenarios.
Let's explore a few everyday examples:

• Distance traveled as a function of time: Whether you're driving or biking, the further you go, the longer it takes. The time spent traveling affects the distance covered.
• Salary as a function of hours worked: In hourly jobs, your total earnings are directly related to the hours you put in. Each extra hour worked results in a proportional increase in your paycheck.
• Electricity bill as a function of electricity used: The amount on your bill depends on your electricity consumption. The more electricity you use, the higher the bill.

By relating these scenarios to functions, we gain insights into how incremental changes in one aspect lead to effects on another. These examples illustrate how functions provide a framework for understanding real-world phenomena.

Domain and Range Concepts

To fully grasp functions, it is crucial to comprehend the ideas of domain and range. The domain consists of all possible input values a function can accept; the range includes all potential outputs the function can produce.
Taking the example of salary as a function of hours worked, the domain would be the number of hours you can work (realistically limited by the hours in a week), and the range would be your possible earnings based on your hourly wage. Each function has its own unique domain and range, dependent on the specific scenario.

• Domain example: For distance as a function of time, the domain involves all the time intervals during the journey.
• Range example: For electricity usage, the range includes all possible bill amounts you could encounter during a billing cycle.

By understanding the domain and range, we can better predict and manage outcomes in various situations. Functions and their domains and ranges are crucial for modeling real-life circumstances, offering clarity and predictability in diverse situations.