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Chapter 5: Problem 32

Determine whether the relation is a function. Explain. $$ (3,4),(4,6),(1,4),(2,-1) $$

Short Answer

Expert verified
Yes, the given relation is a function because each first element (x-value) is associated with exactly one second element (y-value).

Step by step solution

01

Analyze the pairs

Firstly, look at each pair in the set: (3,4),(4,6),(1,4),(2,-1). Observe that each first element in a pair is unique.
02

Identify duplicates

Check to see if there are any first elements that are the same in other pairs. In this case, there are no duplicate first elements.
03

Determine if it is a function

Since each first element in each pair is unique and doesn't repeat, we can conclude that the given relation is a function.

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Key Concepts

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

Relation
In mathematics, a relation is a connection between elements of two sets. To express a relation, we often use ordered pairs, where elements of the first set are associated with elements of the second set. Relations establish what elements are connected in some way. An example of a relation could be pairing students with their respective birth months. Here, the students form one set and the months form another.
  • Relations are not always functions.
  • A relation consists of a set of ordered pairs.
  • Each pair represents an association between two elements.
Understanding how relations work can be pivotal in solving many mathematical problems. Relations help us see connections and patterns in data.
Ordered Pairs
Ordered pairs are fundamental in expressing relations. An ordered pair consists of two elements written in a specific order within parentheses, like \((a, b)\). The first element, "a", is known as the first component, and "b" is the second component. In the context of our exercise, an ordered pair such as \((3, 4)\) suggests that the number 3 is associated with the number 4.

Some key aspects of ordered pairs are:
  • Order matters, \((3, 4)\) is different from \((4, 3)\).
  • They help represent relations in a clear format.
Ordered pairs are not just about numbers; they are used in various fields, including computer science and mapping locations.
Unique First Elements
When analyzing if a relation is a function, one must check if each first element of the ordered pairs is unique. In a function, each input (first element) must map to exactly one output (second element). If any first element repeats with a different second element, the relation isn't a function.

Here is why unique first elements are important:
  • They ensure there is no ambiguity in mapping inputs to outputs.
  • Functions provide a clear, unidirectional relationship.
In our example, the pairs \((3,4),(4,6),(1,4),(2,-1)\), each first element (3, 4, 1, and 2) is different. Therefore, the given relation qualifies as a function.
Mathematics Problem Solving
Problems in mathematics often require identifying whether a relationship is a function. This involves systematic analysis and critical thinking. To determine if a relation is a function, follow these steps:

  • List all ordered pairs.
  • Examine each pair for unique first elements.
  • Check for any repetition of first elements.
  • Confirm that each first element maps to only one second element.
Mastering these steps can significantly aid in solving various mathematical problems. The idea is to simplify complex problems by breaking them into smaller, manageable parts.

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Most popular questions from this chapter

Consider the function \(f(x)=-x\). a. Graph \(f(x)=-x\) and explain why it is its own inverse. Also, verify that \(f(x)=-x\) is its own inverse algebraically. b. Graph other linear functions that are their own inverses. Write equations of the lines you graphed. c. Use your results from part (b) to write a general equation describing the family of linear functions that are their own inverses.

\(f(x)=\sqrt[3]{x^2+10 x}, g(x)=\frac{1}{4} f(-x)+6\)

The surface area \(A\) (in square meters) of a person with a mass of 60 kilograms can be approximated by \(A=0.2195 h^{0.3964}\), where \(h\) is the height (in centimeters) of the person. a. Find the inverse function. Then estimate the height of a 60-kilogram person who has a body surface area of \(1.6\) square meters. b. Verify that function \(A\) and the inverse model in part (a) are inverse functions.

\(f(x)=\sqrt[3]{3 x^2-x}\)

\(x^2+y^2=9\)
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