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Chapter 1: Problem 10

Find all functions (displayed as tables) whose domain is the set {5,8} and whose range is the set {1,3}.

Short Answer

Expert verified
There are 4 possible functions whose domain is the set {5,8} and whose range is the set {1,3}. The functions can be represented by the following tables: Mapping 1: \[ \begin{array}{c|c} \textbf{Domain} & \textbf{Range} \\ \hline 5 & 1 \\ 8 & 1 \\ \end{array} \] Mapping 2: \[ \begin{array}{c|c} \textbf{Domain} & \textbf{Range} \\ \hline 5 & 1 \\ 8 & 3 \\ \end{array} \] Mapping 3: \[ \begin{array}{c|c} \textbf{Domain} & \textbf{Range} \\ \hline 5 & 3 \\ 8 & 1 \\ \end{array} \] Mapping 4: \[ \begin{array}{c|c} \textbf{Domain} & \textbf{Range} \\ \hline 5 & 3 \\ 8 & 3 \\ \end{array} \]

Step by step solution

01

Determine the number of possible mappings

To begin, we will find the number of possible mappings between the two sets. Since there are two elements in the domain set and two elements in the range set, each element in the domain set has two possible outputs in the range set. Thus, there will be a total of 2 x 2 = 4 possible mappings.
02

Create a table for each mapping possible

Now that we know there are 4 possible mappings, let's enumerate them using tables. In each table, the left column (labeled "Domain") will include the elements of the domain set, and the right column (labeled "Range") will include the corresponding output for each mapping.
03

Mapping 1:

\[ \begin{array}{c|c} \textbf{Domain} & \textbf{Range} \\ \hline 5 & 1 \\ 8 & 1 \\ \end{array} \]
04

Mapping 2:

\[ \begin{array}{c|c} \textbf{Domain} & \textbf{Range} \\ \hline 5 & 1 \\ 8 & 3 \\ \end{array} \]
05

Mapping 3:

\[ \begin{array}{c|c} \textbf{Domain} & \textbf{Range} \\ \hline 5 & 3 \\ 8 & 1 \\ \end{array} \]
06

Mapping 4:

\[ \begin{array}{c|c} \textbf{Domain} & \textbf{Range} \\ \hline 5 & 3 \\ 8 & 3 \\ \end{array} \] These are the four possible tables (functions) whose domain is the set {5,8} and whose range is the set {1,3}.

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Key Concepts

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

Understanding the Domain
The domain of a function refers to the set of all possible input values. In other words, it is the collection of values that can be used as arguments for the function.
For the exercise we're discussing, the domain is the set \( \{5,8\} \). This means that our function can only accept 5 or 8 as inputs from this set.
Understanding the domain is crucial because it defines what is permissible as input to the function.
  • If the domain includes the number 5, the function must provide an output for 5 based on the range.
  • Similarly, if the 8 is part of the domain, an output for 8 should exist in the function.
Being clear on the domain helps to avoid errors when working with functions and to know exactly what values you can plug into the function without causing any disruptions.
Unveiling the Range
The range of a function is the set of all possible outputs, the values you can get after applying the function rule to every element of the domain.
In the exercise, the range is set to be \( \{1,3\} \). This means that any function constructed from this example should only provide outputs of 1 or 3.
  • The range captures the fundamental idea of what the function does to the input from the domain.
  • It is a crucial part of defining the characteristics and behavior of a function.
Understanding the range allows you to anticipate what results you can expect from the function when you enter an element from the domain.
Exploring Mappings
Mappings in the context of functions refer to the specific pairing or association of each element from the domain to an element in the range.
For the given exercise, there are four possible mappings that pair each number in the domain \( \{5,8\} \) with numbers in the range \( \{1,3\} \).
Mapping is a way to illustrate the function visually or through a table, making it easier to understand.

Here are the key steps regarding mappings:
  • Identify all elements in the domain and the range.
  • Pair each domain value with every range value through logical associations.
  • Make sure that every domain value has a corresponding output (this forms a valid function).
By understanding the concept of mappings, you can easily determine how inputs from the domain lead to specific outputs in the range, emphasizing the function's internal logic.

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Most popular questions from this chapter

For each of the functions \(f\) given in Exercises \(13-\) 22: (a) Find the domain of \(f\). (b) Find the range of \(f\). (c) Find a formula for \(f^{-1}\). (d) Find the domain of \(\boldsymbol{f}^{-1}\). (e) Find the range of \(f^{-1}\). You can check your solutions to part (c) by verify. ing that \(f^{-1} \circ f=I\) and \(f \circ f^{-1}=I\) (recall that \(I\) is the function defined by \(I(x)=x\). $$ f(x)=\frac{1}{3 x+2} $$

Suppose \(f\) and \(g\) are functions, each of whose domain consists of four numbers, with \(f\) and \(g\) defined by the tables below: $$ \begin{array}{c|c} {x} & {f}({x}) \\ \hline {1} & 4 \\ 2 & 5 \\ 3 & 2 \\ 4 & 3 \end{array} $$ $$ \begin{array}{c|c} x & g(x) \\ \hline 2 & 3 \\ 3 & 2 \\ 4 & 4 \\ 5 & 1 \end{array} $$ Sketch the graph of \(f^{-1}\).

(a) True or false: The product of an even function and an odd function (with the same domain) is an odd function. (b) Explain your answer to part (a). This means that if the answer is "true", then explain why the product of every even function and every odd function (with the same domain) is an odd function; if the answer is "false", then give an example of an even function \(f\) and an odd function \(g\) (with the same domain) such that \(f g\) is not an odd function.

Show that if \(f\) is an odd function such that 0 is in the domain of \(f\), then \(f(0)=0\).

A constant function is a function whose value is the same at every number in its domain. For example, the function \(f\) defined by \(f(x)=4\) for every number \(x\) is a constant function. Give an example of three functions \(f, g,\) and \(h\), none of which is a constant function, such that \(f \circ h=g \circ h\) but \(f\) is not equal to \(g\).
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