Question

Consider the set \(f=\\{(x, y) \in \mathbb{Z} \times \mathbb{Z}: 3 x+y=4\\} .\) Is this a function from \(\mathbb{Z}\) to \(\mathbb{Z} ?\) Explain.

Short answer

Yes, the given set \(f\) is a function from \(\mathbb{Z}\) to \(\mathbb{Z}\).

Step-by-step solution

Identify the function

The given set is \(f=\{(x, y) \in \mathbb{Z} \times \mathbb{Z}: 3 x+y=4\}\). This relation says that for every number \(x\) in \(\mathbb{Z}\), \(y\) would be \(4 - 3x\), also a number in \(\mathbb{Z}\).

Test for function properties

For the set \(f\) to be a function from \(\mathbb{Z}\) to \(\mathbb{Z}\), every \(x\) in \(\mathbb{Z}\) must result in only one \(y\) value in \(\mathbb{Z}\). Check an example: for \(x = 1\), the equation yields \(y = 1\). Using another integer \(x = 2\), the equation yields \(y = -2\). Thus, for every single value of \(x\), there is a single corresponding value of \(y\).

Determine if the set \(f\) is a function

Since for every value of \(x\) in \(\mathbb{Z}\) there is exactly one corresponding value of \(y\) in \(\mathbb{Z}\), the set \(f\) satisfies the definition of a function from \(\mathbb{Z}\) to \(\mathbb{Z}\).

Key concepts

Functions in Mathematics

In mathematics, a function is a relation between two sets that assigns to each element of the first set (the domain) exactly one element of the second set (the co-domain). Why does this matter? Functions are like secret codes that give you specific outputs when you input certain numbers – like a vending machine that gives you exactly one type of snack when you enter a code. Using the equation from our textbook exercise, where for every number x in \(\mathbb{Z}\), y would be \(4 - 3x\), we can see each input x gives us one output y. Whether you plug in 1 or 100, you'll get a corresponding single y.

Understanding functions is a cornerstone of algebra and helps build a foundation for more advanced mathematics. They describe many real-world phenomena and allow us to predict and understand the world in a systematic way.

Real-World Application

For instance, functions can represent the relationship between the distance traveled and the time taken, or the temperature change over the course of a day.

Function Properties

A function's properties tell us how it behaves and what it can do. The function we're looking at is defined by an equation \(3x + y = 4\). One key property is that it should give us one and only one output for each input. It's sort of like saying, in a perfect world – for each work you do, there's only one paycheck amount you'd get.

Other properties include continuity, where a function has no abrupt changes; domain and co-domain, which are the sets that the function’s inputs and outputs belong to; and range, the set of all possible outputs given the inputs. The beauty of functions is that once you know their properties, you can predict and calculate outcomes, analyze patterns, and make sense of data.

• One-to-One: Every element of the domain is paired with a unique element from the co-domain.
• Onto: Each element in the co-domain is an output of the function.
• Injective: Each output comes from only one input.

Relation Between Sets

When we talk about functions, we're essentially discussing a special type of relation between sets. Not all relations are functions, but every function is a relation. Let's take the example in our exercise f, where we have a set of ordered pairs that satisfy the equation \(3x + y = 4\).

A function is a well-behaved relation that makes sure no member of the domain (the first set, or in our case, all integers \(\mathbb{Z}\)) is feeling neglected – it’s guaranteed to have a partner in the co-domain (the second set). As seen in steps to solve our exercise, the relation f passes the test of being a function, since every x from the set \(\mathbb{Z}\) snuggles up with exactly one y from \(\mathbb{Z}\). This shows the close relationship between the concept of functions and the broader concept of relations between sets.