/*! 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 8 For Problems 7 through 9 determi... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 8

For Problems 7 through 9 determine whether the relationship described is a function. If the relationship is a function, (a) what is the domain? the range? (b) is the function 1 -to- 1 ? $$ \begin{array}{ll} \text { Input } & \text { Output } \\ \hline \sqrt{2} & 2 \\ \sqrt{3} & 3 \\ \sqrt{5} & 5 \\ \sqrt{6} & 6 \end{array} $$

Short Answer

Expert verified
The relationship described is indeed a function. The domain of this function is \(\sqrt{2}\), \(\sqrt{3}\), \(\sqrt{5}\), \(\sqrt{6}\), and the range is 2, 3, 5, 6. The function is a 1-to-1.

Step by step solution

01

Understand the definition of a function

A relationship between two quantities is defined as a function if every input has exactly one output. Looking at the problem, we see that every input (\(\sqrt{2}\), \(\sqrt{3}\), \(\sqrt{5}\), \(\sqrt{6}\)) has exactly one corresponding output (2, 3, 5, 6), so it is a function.
02

Identify the domain of the function

The domain of a function is the set of all possible input values. In this case the domain is the set of all inputs, so the domain is \(\sqrt{2}\), \(\sqrt{3}\), \(\sqrt{5}\), \(\sqrt{6}\).
03

Identify the range of the function

The range of a function is the set of all possible output values. Looking at the problem, we see that the range is the set of all outputs, so the range is 2, 3, 5, 6.
04

Determine if the function is 1-to-1

A function is described as 1-to-1, or injective, if every output is generated by exactly one input. Here, no two inputs map to the same output, so the function is 1-to-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.

Domain
In mathematics, when discussing functions, the **domain** is an essential concept. The domain refers to the complete set of possible input values that a function can accept. Understanding the domain is crucial because it outlines the "inputs" for the function. Features of the domain include:
  • The domain consists solely of inputs that will not lead to undefined situations in the function.
  • Typically represented using set notation or listing actual values when possible.
  • Sometimes restrictions on domains occur due to mathematical operations like division by zero or taking square roots of negative numbers.
In our problem, the domain comprises the specific values:
  • \( \sqrt{2} \)
  • \( \sqrt{3} \)
  • \( \sqrt{5} \)
  • \( \sqrt{6} \)
Each of these inputs is a unique non-negative square root of a prime or composite number, suitable for the function described in the problem.
Range
While the domain is all about inputs, the **range** relates to the outputs of a function. In simpler terms, it鈥檚 the set of all possible results that a function can produce from its domain. Here are some essential points:
  • Determining the range requires observing the outcomes derived from each input within the domain.
  • It often requires solving the function for each possible input or understanding the logic or pattern of the outputs.
  • Writing the range usually involves listing all possible outcome values that the function might yield.
In the given situation, the range of this function consists of the outputs:
  • 2
  • 3
  • 5
  • 6
Each of these numbers corresponds directly to one of our domain values. Since each input matches a unique output, determining the range is straightforward in this scenario.
Injective Function
An **injective function**, also known as a one-to-one function, is a type of function where each output value is uniquely tied to one input. Injective functions are crucial because they guarantee that information is not lost when transitioning from input values to output values. Here鈥檚 what you need to know:
  • Each element in the range is mapped by exactly one element in the domain.
  • If different inputs lead to the same output, the function is not injective.
  • Injective functions simplify reversing the mapping process since each output corresponds to a unique input.
In the given example, every output from the function (\(2, 3, 5, 6\)) is determined by a unique input (\(\sqrt{2}, \sqrt{3}, \sqrt{5}, \sqrt{6}\)). Consequently, the function can be described as injective or one-to-one. This property ensures that each input value corresponds to a distinct output value without any overlap or repetition.

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

Express the surface area of a cube as a function of the length \(s\) of one side.

A gardener has a fixed length of fence that she will use to fence off a rectangular chili pepper garden. Express the area of the garden as a function of the length of one side of the garden. If you have trouble, reread the "Portable Strategies for Problem Solving" listed in this chapter. We've also included the following advice geared specifically toward this particular problem. Give the length of fencing a name, such as \(L\). (We don't know what \(L\) is, but we know that it is fixed, so \(L\) is a constant, not a variable.) \- Draw a picture of the garden. Call the length of one side of the fence \(s\). How can you express the length of the adjacent side in terms of \(L\) and \(s ?\) \- What expression gives the area enclosed by the fence?

If \(f(x)=\sqrt{\frac{1}{x+1}}\), find the following. Simplify your answer where possible. (a) \(f(0)\) (b) \(f(3)\) (c) \(f\left(-\frac{1}{4}\right)\) (d) \(f(b)\) (e) \(f(b-1)\) (f) \(f(b+3)\) (g) \([f(7)]^{2}\) (h) \(f\left(b^{2}\right)\) (i) \([f(b)]^{2}\)

At the Central Perk coffeehouse in Manhattan, Rachel serves \(c\) cups of coffee and \(d\) desserts per hour. The coffee costs \(a\) dollars per cup, and the desserts cost \(b\) dollars each. She averages a tip of \(k\) cents per dollar of the customers' bills (excluding taxes). In addition, she makes a fixed wage of \(F\) dollars per hour. Consider \(c, d, a, b, k\), and \(F\) as constants. Express Rachel's earnings as a function of \(h\), the number of hours she works. (In actuality, Rachel's earnings are not a function of the hours she puts in. Other considerations complicate the situation. For instance, business is slow at certain times of the day, and some customers tip more generously than others. Nevertheless, by using the information provided, we can make a mathematical model of the situation that gives us a reasonably accurate picture.)

A rectangle is inscribed in a semicircle of radius \(R\), where \(R\) is constant. (a) Express the area of the rectangle as a function of the height, \(h\), of the rectang \(A=f(h)\) (b) Express the perimeter of the rectangle as a function of the height, \(h\), of rectangle, \(P=g(h)\).
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Applied Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Calculus

Read Explanation

Discrete Mathematics

Read Explanation

Theoretical and Mathematical Physics

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.