Question

Crimes Suppose that \(V(x)\) represents the number of violent crimes committed in year \(x\) and \(P(x)\) represents the number of property crimes committed in year \(x\). Determine a function \(T\) that represents the combined total of violent crimes and property crimes in year \(x\).

Short answer

The function is \ T(x) = V(x) + P(x) \.

Step-by-step solution

Understand the Problem Statement

Given two functions, one representing violent crimes and the other representing property crimes in the same year, combine these functions to represent the total crimes.

Define the Given Functions

Let the function for violent crimes be represented by \(V(x)\) and the function for property crimes be represented by \(P(x)\).

Combine the Functions

To find the total number of crimes in year \(x\), add the functions for violent and property crimes together. This can be written as: \ T(x) = V(x) + P(x) \.

Write the Final Function

The final function that represents the combined total of violent and property crimes in year \(x\) is: \ T(x) = V(x) + P(x) \.

Key concepts

addition of functions

The addition of functions is a way to combine two or more functions into one. In this exercise, we are combining the function that represents violent crimes, denoted as \(V(x)\), with the function that represents property crimes, denoted as \(P(x)\).

To add two functions, you simply sum up their values for the same input.
Formulaically, if you have two functions \(f(x)\) and \(g(x)\), their addition is given by:
\[ (f + g)(x) = f(x) + g(x) \]
In this case, the total crimes function \(T(x)\) is:
\[T(x) = V(x) + P(x) \]
This formula means that for any given year \(x\), the value of \(T(x)\) is found by adding the number of violent crimes and property crimes committed that year.

total crimes function

The total crimes function, \(T(x)\), represents the combined total of violent and property crimes committed in a specific year, \(x\). This function helps to gain an overall understanding of crime rates by summing different categories of crimes.

Given:

• \(V(x)\): Number of violent crimes in year \(x\)
• \(P(x)\): Number of property crimes in year \(x\)

The total crimes function is derived by adding these two values together:
\[ T(x) = V(x) + P(x) \]
This new function \(T(x)\) represents the total number of crimes in year \(x\). This makes it easy to assess the overall crime situation in that year.

Breaking crimes into different types and then combining them into one function is a common technique in statistics and data analysis. It helps in simplifying complex information and making it more digestible.

violent and property crimes

Understanding the specifics of violent and property crimes is crucial for interpreting the total crimes function.

• **Violent Crimes (\(V(x)\))**: These include actions that cause or threaten physical harm, such as assault, robbery, and murder. These are often considered more serious due to the direct threat to personal safety.
• **Property Crimes (\(P(x)\))**: These involve the theft or destruction of property but do not generally pose a direct threat to personal safety. Examples include burglary, larceny, and motor vehicle theft.

Combining these into a single function, \(T(x)\), provides a more comprehensive overview of the total crime rate. It’s a holistic approach to understanding the crime scene, allowing authorities to allocate resources and plan strategies more effectively.

This exercise teaches the mathematical concept of adding functions while also stressing the importance of collaborative data for informed decision-making.