Функції: відношення

Relations vs. Functions

A relation is just a relationship between $x$ and $y$-coordinates. It maps inputs to outputs. $$\text{example:}\quad y^2=x$$

Notice that, given an input of $x,$ there can be multiple outputs of $y$ that satisfy the relation represented by this equation. For example, if $x=4,$ then $y$ can be $2$ or $-2.$

In the set of all relations, there's a smaller subset which we call functions. A function is just a special kind of relation: any input has exactly one output. $$\text{example:}\quad y=x+3$$

Here, there is only one output $y$ per input $x.$ For example, if $x=4,$ then $y$ can only be $7.$

Is it a Function?

So how do we tell if something if a function or just a relation? As touched upon above, we look at the inputs and outputs.

There are $4$ different relations we can come across: one-to-one, one-to-many, many-to-one, and many-to-many.

• One-to-one: Each $x$-value corresponds to one distinct $y$-value.
• One-to-many: One $x$-value corresponds to multiple $y$-values.
• Many-to-one: Multiple $x$-values correspond to the same $y$-value.
• Many-to-many: Multiple $x$-values correspond to multiple $y$-values.

A relation is a function if the $x$-values map to only one $y$-value. In other words, if a relation is one-to-one or many-to-one it is a function. $$\begin{array}{c|c} \text{Function}&\text{Not a Function}\\ \hline \text{one-to-one}&\text{many-to-many}\\ \text{many-to-one}&\text{one-to-many} \end{array}$$

Example 1

Let's figure out what kind of relation the following is and if it's also a function. $$y=3x^2-3$$

Start by plugging in a few values for $x$ and making a table. $$\begin{array}{c|c} x&y\\ \hline -2&9\\ -1&0\\ 0&-3\\ 1&0\\ 2&9 \end{array}$$

From here we can already find what kind of relation this is, but it's easier to visualize by mapping so let's do that.

Each input only has one output. However, many inputs have the same output. So, this is a many-to-one function.

Graphically

Graphically, we can easily tell if a relation is a function using the vertical line test. This will be covered in more detail in a later section.

Solution

Let's map these coordinates and see.

It's a bit of a jumbled mess, but the $x$ side has multiple lines coming out of $3$. This means the $x$ side is many. Similarly, the $y$ side has multiple lines going to $5$. This is also many.

Thus, our relation is many-to-many.

Solution

To tell if this is a function, just look at the $x$-values. In the mapping above, our relation has multiple lines coming from $x=3$. This means for that input there's more than one output: $y=1$ and $y=4$. A function can only have one output per input. Therefore, this relation is not a function.

We'll use this concept of equations being functions later, when we're solving linear functions.

When is a Relation a Function?

Recall that a relation can map inputs to multiple outputs. It is a function when it maps each input to exactly one output. The set of all functions is a subset of the set of all relations. That means all functions are relations, but not all relations are functions.

Consider $y^2=x.$ This equation represents a relation that is not a function. Why? Because each nonzero input $x$ maps to two outputs $y.$ For example, if $x=9,$ then $y=\pm 3.$

Now, consider $y=2x.$ This equation represents a function (and so also a relation). Why? Because each nonzero input $x$ maps to exactly one output $y.$ For example, if $x=9,$ then $y=18.$

You can use the vertical line test to check if you are graphing a function or not!

(Video) Determining if a Relationship is a Function

by HCCMathHelp

Here's a video by HCCMathHelp that goes over how you can tell when a relation is a function.

$\large{\text{Summary}}$

Recall that a function is a rule where for any given input, it gives one specific output.

For example, $y=x+2$ is a function. For any input $x$, you get out a $y$. When $x=3$, the function always gives the output $y=5$.

Tables

Example 1$\\$ Relations and functions can also be expressed using tables of values. For example: $$\begin{array}{c|c} x&y\\ \hline -1&3\\ 0&5\\ 1&7\\ 2&9 \end{array}$$ The $x$ column is the inputs, and the $y$ column is the outputs.

So when you input a $-1$ you get an output of $3$. When you input $0$ you get an output of $5$. And so on. This is a function because whatever we input, we know exactly what we'll get as an output.

Example 2$\\$ Let's now look at an example where this isn't the case. $$\begin{array}{c|c} x&y\\ \hline -1&3\\ 0&5\\ -1&7\\ 2&9 \end{array}$$ Here our input of $-1$ gives us two different outputs: either $y=3$ or $y=7$. This is not a function. But it is still a relation.

Example 3$\\$ What if the $y$-values repeat. For example: $$\begin{array}{c|c} x&y\\ \hline -1&5\\ 0&6\\ 1&7\\ 2&6 \end{array}$$ Even though different inputs lead to the same output, it's still a function. We have one specific output we know we'll get when we put in the input.

In general, as long as the $x$-values don't repeat, we will have a function!

Set of Ordered Pairs

Example 1$\\$ Sometimes instead of tables, there'll bet sets of ordered pairs. For example: $$\{(3,5),(4,7),(5,9),(6,11)\}$$

To tell if this is a function, we just need to look at the $x$-values. Do any of them repeat? $$\{({\color{#8925ae}{3}},5),({\color{#8925ae}{4}},7),({\color{#8925ae}{5}},9),({\color{#8925ae}{6}},11)\}$$ No. Therefore, this is a function.

Example 2$\\$ What about this set? $$\{(3,5),(4,8),(3,7),(4,9)\}$$ Here, we have a repeat of $3$ for the $x$'s. Therefore, this is not a function.

Example 3$\\$ Finally, what about this one? $$\{(3,6),(4,6),(5,6),(6,6)\}$$ Even though the $y$-values repeat, a distinct $x$-value goes to each of them. This is still a function.

Graphs

You can easily tell if a graph is a function by using the vertical line test. This will be covered in more detail in a later section.

The basic gist is if a vertical line touches more than one point at any time on your graph, the graph is not a function.

Equations

As long as an $x$-value doesn't give multiple $y$-values, the equation will be a function.

Example 1$\\$ The equation $y=-2x+4$ only gives one $y$-value for any $x$-value. Therefore, it is a function.

Example 2$\\$ The equation $y=x^2-4x+3$ only gives one $y$-value for any $x$-value. The $y$-values may repeat, but that's okay. Therefore, this is a function.

Example 3$\\$ The equation $y=\pm x$ is not a function. The plus/minus symbol, $\pm$, means our $x$ input will either give out plus that amount or minus that amount. If $x=2$, then the output is either $y=2$ or $y=-2$.