Problem 33
Question
Determine the domain \(D\) and range \(R\) of each relation, and tell whether the relation is a function. Assume that a calculator graph extends indefinitely and a table includes only the points shown. $$\begin{array}{|c|c|c|c|c|c|} \hline x & 0 & 1 & 2 & 3 & 4 \\ \hline y & \sqrt{2} & \sqrt{3} & \sqrt{5} & \sqrt{6} & \sqrt{7} \end{array}$$
Step-by-Step Solution
Verified Answer
Domain: \( \{ 0, 1, 2, 3, 4 \} \), Range: \( \{ \sqrt{2}, \sqrt{3}, \sqrt{5}, \sqrt{6}, \sqrt{7} \} \); The relation is a function.
1Step 1: Identify the Domain
The domain of a relation includes all possible input values, which are the first row of the table. From the table, the domain \( D \) is the set of x-values: \( \{ 0, 1, 2, 3, 4 \} \).
2Step 2: Identify the Range
The range of a relation includes all possible output values, which are the second row of the table. From the table, the range \( R \) is the set of y-values: \( \{ \sqrt{2}, \sqrt{3}, \sqrt{5}, \sqrt{6}, \sqrt{7} \} \).
3Step 3: Determine if the Relation is a Function
A relation is a function if each input (x-value) is related to exactly one output (y-value). In the given table, each \( x \)-value corresponds to a unique \( y \)-value with no repetitions. Thus, the relation satisfies the definition of a function.
Key Concepts
Domain and RangeRelationsInput and Output Values
Domain and Range
Whenever we talk about functions and relations, the concepts of domain and range are essential. The domain, simply put, is the complete set of possible input values, typically represented by the variable \( x \). In the context of a table, or any function, these values are usually listed in the first row or column. For the example, the domain is \( \{0, 1, 2, 3, 4\} \), which are the \( x \)-values.
On the other hand, the range is the complete set of possible output values. These are typically represented by the variable \( y \) and are usually found in the second row or column of a table. For the given relation, the range is \( \{ \sqrt{2}, \sqrt{3}, \sqrt{5}, \sqrt{6}, \sqrt{7} \}\).
Understanding the domain and range helps us fully describe what a relation or function can potentially look like, clarifying both its inputs and possible outcomes.
On the other hand, the range is the complete set of possible output values. These are typically represented by the variable \( y \) and are usually found in the second row or column of a table. For the given relation, the range is \( \{ \sqrt{2}, \sqrt{3}, \sqrt{5}, \sqrt{6}, \sqrt{7} \}\).
Understanding the domain and range helps us fully describe what a relation or function can potentially look like, clarifying both its inputs and possible outcomes.
Relations
A relation in mathematics refers to the connection between a set of inputs and a set of possible outputs. Think of it like a rule that links \( x \)-values (inputs) to \( y \)-values (outputs).
Relations are visible in various forms – tables, graphs, sets of ordered pairs, and mappings. In our example, the relation is given in a table format, which precisely shows which \( x \)-value is paired with which \( y \)-value.
Analyzing these connections allows us to determine several important features, such as the existence of repeated outputs or inputs, which are vital in classifying a relation as a function.
Relations are visible in various forms – tables, graphs, sets of ordered pairs, and mappings. In our example, the relation is given in a table format, which precisely shows which \( x \)-value is paired with which \( y \)-value.
Analyzing these connections allows us to determine several important features, such as the existence of repeated outputs or inputs, which are vital in classifying a relation as a function.
Input and Output Values
In any function or relation, we center our analysis on input and output values. Inputs, designated as \( x \)-values, are the values we put into a function or relation. In simpler terms, they are the questions we ask.
Outputs, represented as \( y \)-values, are the results or answers the relation provides based on the inputs. In our table, inputs are \( \{0, 1, 2, 3, 4\} \) and each input corresponds uniquely to outputs \( \{ \sqrt{2}, \sqrt{3}, \sqrt{5}, \sqrt{6}, \sqrt{7} \} \).
This direct pairing is essential for understanding if the relation is a function. In a function, each input should pair with exactly one output, ensuring no input is left without a clear, single answer. This is evident in our relation, confirming it as a function.
Outputs, represented as \( y \)-values, are the results or answers the relation provides based on the inputs. In our table, inputs are \( \{0, 1, 2, 3, 4\} \) and each input corresponds uniquely to outputs \( \{ \sqrt{2}, \sqrt{3}, \sqrt{5}, \sqrt{6}, \sqrt{7} \} \).
This direct pairing is essential for understanding if the relation is a function. In a function, each input should pair with exactly one output, ensuring no input is left without a clear, single answer. This is evident in our relation, confirming it as a function.
Other exercises in this chapter
Problem 33
A line having an equation of the form \(y=k x\), where \(k\) is a real number, \(k \neq 0\), will always pass through the origin \((0,0) .\) To graph such an eq
View solution Problem 33
Give the equation of the \(x\) -axis.
View solution Problem 33
do each of the following. (a) Express the cost \(C\) as a function of \(x,\) where \(x\) represents the number of items as described. (b) Express the revenue \(
View solution Problem 33
Locate each point on a rectangular coordinate system. Identify the quadrant, if any, in which each point lies. $$(3,0)$$
View solution