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Chapter 2: Problem 26

Determine whether each relation is a function. $$\\{(1, \pi),(30, \pi / 2),(60, \pi / 4)\\}$$

Short Answer

Expert verified
The relation is a function because every input has one unique output.

Step by step solution

01

Identify the Pairs

List out all the pairs in the given relation: \((1, \pi), (30, \pi/2), (60, \pi/4)\).
02

Check for Unique Inputs

Verify that each input (the first number in each pair) appears only once. The inputs here are \(1, 30, 60\).
03

Analyze the Outputs for Given Inputs

Check if every input has exactly one output. For the inputs \(1, 30, 60\), the outputs are \(\pi, \pi/2, \pi/4\) respectively.
04

Conclude If the Given Relation is a Function

Since each input has a unique output, 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.

Relations and Functions
In mathematics, a relation is a collection of ordered pairs. These pairs link an input to an output, kind of like a matching game. For example, consider the pairs \((1, \pi), (30, \pi/2), (60, \pi/4)\). Here, each pair connects an input value to an output value.
But what makes a relation a function?
A function is a special type of relation. It requires that every input (often called the 'domain') has exactly one output (often called the 'range').
This means that if you look at each first number in the pairs, none of them should repeat. If they do, it's not a function. This uniqueness makes the connection predictable and reliable, fulfilling the criteria of a function.
Unique Inputs
Now, let's dive into what makes inputs unique and why it is essential for defining functions.
Unique inputs mean that within our ordered pairs, each first value (input) must be different.
In our example, the inputs are 1, 30, 60. Notice how none of these numbers repeat? That鈥檚 good! The uniqueness of inputs ensures that each input is only linked to one output.
Imagine if you got the input 1 in two pairs, like \((1, \pi)\) and \((1, \pi/3)\). This would confuse the function's definition, as one input would give two possible outputs. Therefore, all inputs must be different to maintain the integrity of the function.
Output Analysis
After verifying the uniqueness of the inputs, the next step involves examining the outputs. We need to check that each input is tied to one and only one output.
Let's look at our existing relation: \((1, \pi)\), \((30, \pi/2)\), \((60, \pi/4)\). The outputs here are \pi, \pi/2, and \pi/4 respectively.
Each input matches a single output uniquely. This single mapping is crucial.
It wouldn't qualify as a function if any input matched multiple outputs. For example, if one input pointed to both \pi and \pi/2, it would violate the rule. Therefore, analyzing outputs for uniqueness is a key step in determining if a relation is indeed a function.

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