Problem 26 - Algebra Form and Function

Question

Put the functions in the form \(Q=k t\) and state the value of \(k\). $$ Q=t(n+1) $$

Step-by-Step Solution

Answer

Answer: The constant \(k\) is equal to \((n+1)\).

Step 1: Identify Variables and Constants

In the given function \(Q = t(n + 1)\), we can see that \(Q\) and \(t\) are variables, while \(n\) is a constant.

Step 2: Rewrite the Function

To rewrite the function in the form \(Q = kt\), we need to find a way to express \(n+1\) so that it is constant. Since \(n\) is a constant, its addition to 1 would still result in a constant. In other words, \((n+1)\) can be represented as a single constant, which we will call \(k\). So, the function can now be written as: $$Q = kt$$

Step 3: Determine the Value of k

As we established in Step 2, the constant \(k\) is equal to \((n+1)\). Therefore, the value of \(k\) is: $$k = n + 1$$

Key Concepts

Variables

In mathematics, a variable is essentially a symbol that represents a number or a value that can change. It can take on different values in different situations or problems. In the expression given in the exercise, both \(Q\) and \(t\) are variables, meaning they do not hold a fixed value and can vary depending on the context or input they're given.
Variables are crucial in helping us create general formulas and functions that can be applied to a wide range of situations. They allow for flexibility and adaptability in mathematics by acting as placeholders for unknowns or quantities that can change.

• Example: In \(Q = t(n+1)\), both \(Q\) and \(t\) are variables, which means they can take on any numerical value depending on what is being tested or observed.
• Importance: Understanding variables helps in solving equations and modeling real-life scenarios mathematically.

Constants

A constant, unlike a variable, is a fixed value. In equations or functions, constants are numbers or symbols that do not change. Their value remains the same throughout the problem-solving process.
In the exercise's function \(Q = t(n+1)\), \(n+1\) functions as a constant because \(n\) is already a fixed number, and adding \(1\) doesn't change the status of being constant.

• Role of Constants: Constants often serve as specific coefficients that provide a baseline or scale to a mathematical function or equation.
• Example: In a classroom setting, if \(n\) represents the number of pencils each student receives and remains the same, then \(n+1\), which might represent the total pencils including one extra, is a constant.
• Conclusion: Recognizing constants helps simplify equations and identify the essence of a function.

Function Notation

Function notation is a way of representing functions in a mathematical format, making them easier to recognize and work with. The use of letters or symbols like \(f(t)\) or \(Q=f(t)\) conveys the idea that \(Q\) is a result of the function \(f\) applied to \(t\).
Using clear and precise notation is important as it helps in describing mathematical operations clearly. It presents an organized way to express relationships between variables and constants.

• Notation: In the problem, rewriting \(Q = t(n+1)\) to \(Q = kt\) using function notation clarifies that \(Q\) depends on \(t\).
• Purpose: Function notation simplifies the complex expressions and indicates the dependency and interaction between variables.
• Example: If we said \(f(t) = kt\), this explicitly shows function \(f\) acting on \(t\), defined by a constant \(k\).