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Chapter 1: Problem 12

Determine whether each relation is a function. Assume that the coordinate pair \((x, y)\) represents the independent variable \(x\) and the dependent variable \(y .\) $$y=3$$

Short Answer

Expert verified
The relation described by \( y = 3 \) is a function because each x-value maps to the same y-value.

Step by step solution

01

Understanding Relations

A relation is a set of ordered pairs, where each pair consists of an input (independent variable) and an output (dependent variable). In our case, we consider pairs of \(x, y\).
02

Defining a Function

A relation is defined as a function if each input value (x) corresponds to exactly one output value (y). This means no x-value is paired with more than one y-value.
03

Analyzing the Given Equation

The given equation is \(y=3\). This equation implies that for any value of x, the output y is always 3.
04

Assessing Each Input and Output Pair

Since the equation \(y=3\) holds for any x, for every input x, we get the output pair \(x, 3\). This implies one unique y-value (3) for every x-value.
05

Concluding the Function Status

Because every x-value maps to exactly one y-value (3), the relation described by \(y=3\) is a function.

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Key Concepts

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

Relations
In mathematics, a **relation** is an association between two sets, usually called the domain and the range. These sets are made up of elements, and when discussing functions, we typically consider relations among coordinate pairs like \((x, y)\), where \(x\) is the input and \(y\) is the output. The collection of these pairs makes up the relation.
A relation does not necessarily mean each input has only one output. Unlike functions, relations can have multiple outputs for a single input.
  • For example, in the relation \(R = \{(1, 2), (1, 3), (2, 4)\}\), the input 1 is associated with both outputs 2 and 3.
  • This means \(R\) is not a function.
However, if for every element \(x\) there is only one element \(y\), then the relation is a function.
Independent Variable
The **independent variable** is the variable that stands alone and isn't changed by the other variables you are trying to measure. In math problems like the one in our original exercise, it's the variable you're free to pick values for. This freedom makes it independent.
Usually, we denote the independent variable by \(x\) in equations or functions.
  • In a function, the independent variable is the 'cause' or 'input'.
  • It controls how the dependent variable changes.
In the equation \(y = 3\), \(x\) takes on any value, while \(y\) remains constant at 3.
Dependent Variable
The **dependent variable** is what you measure or the result that depends on what has been changed in the independent variable. It's the output or the 'effect' that relies on the input value of the independent variable.
When we use the term dependent variable, we are generally talking about the variable represented by \(y\) in functions.
  • The dependent variable varies according to the input value of \(x\).
  • Its value is determined by the rule set by the function or equation.
In our specific example from the exercise, the equation \(y = 3\) indicates the dependent variable \(y\) does not depend on \(x\); it is always 3, no matter the value of \(x\). This makes it a constant function and further reinforces the concept of a function mapping a constant output to every possible input value.

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