/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 26 Put the functions in the form \(... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 26

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

Short Answer

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

Step by step solution

01

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.
02

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: $$Q = kt$$
03

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: $$k = n + 1$$

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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 \(k\).

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Are the two functions the same function? $$ A(n)=(n-1) / 2 \text { and } B(n)=0.5 n-0.5 $$

Let \(r(p)\) be the revenue in dollars that a company receives when it charges \(p\) dollars for a product. Explain the meaning of the following statements. (a) \(r(15)=112,500\) (b) \(\quad r(a)=0\) (c) \(r(1)=b\) (d) \(c=r(p)\)

Are the two functions the same function? $$ r(x)=5(x-2)+3 \text { and } s(x)=5 x+7 $$

Abby and Leah go on a 5 hour drive for 325 miles at 65 mph. After \(t\) hours, Abby calculates the distance remaining by subtracting \(65 t\) from \(325,\) whereas Leah subtracts \(t\) from 5 then multiplies by \(65 .\) (a) Write expressions for each calculation. (b) Do the expressions in (a) define the same function?

Table 4.3 shows the 5 top winning teams in the NBA playoffs between 2000 and May 20,2007 and the number of games each team has won. $$ \begin{array}{c|c} \hline \text { Team } & \text { Playoff games won } \\ \hline \text { Lakers } & 66 \\ \hline \text { Spurs } & 66 \\ \hline \text { Pistons } & 61 \\ \hline \text { Nets } & 43 \\ \hline \text { Mavericks } & 41 \\ \hline \end{array} $$ (a) Is the number of games a team won a function of the team? Why or why not? (b) Is the NBA team a function of the number of games won? Why or why not?
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Geometry

Read Explanation

Applied Mathematics

Read Explanation

Calculus

Read Explanation

Mechanics Maths

Read Explanation

Probability and Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.