Introduction to Relations and Functions (L9.1)

Welcome to Introduction to Relations and Functions.

Let's begin with two definitions.

A relation is any set of ordered pairs.

A function is a relation in which every input value is paired with exactly one output value.

So a function is a relation, but it's a special relation in which every input value is paired with exactly one output value.

And because the input variable is often x and the output variable is often y, sometimes we may hear the definition of a function as a relation in which every x-value is paired with exactly one y-value.

One way to represent the relationship between the input and output variables in a relation or function is by the means of a table of values which we see here below.

Notice how we have a column of inputs and a column of outputs.

We're asked to determine which of the following tables represent functions.

So by analyzing the table, we're going to determine if every input value is paired with exactly one output value.

If it is, it's a function.

But if we have an input that is paired with more than one output value, it's not a function.

It's still a relation, but not a function.

Looking at our first table, notice how one is paired with five, two is paired with five, three is paired with five, and four is paired with five.

Notice how every input is paired with exactly one output.

Even though they're all paired to the same output, that doesn't matter.

Every input is paired with exactly one output.

So the answer is yes, this is a function.

In the next table, one is paired with eight, two is paired with negative nine, three is paired with seven, and three is paired with 12.

This table has a problem.

Notice how the input of three is not paired with exactly one output, it's paired with two outputs.

Three is paired with seven and three is also paired with 12.

Therefore, this is not a function.

It's still a relation, but it's not a function, so we say no.

In our third table, two is paired with four, one is paired with negative five, four is paired with 10, and negative three is paired with negative 87.

Once again, notice how every input is paired with exactly one output.

Therefore, this is a function, so we say yes.

Going back to this table that was not a function, notice how the problem occurs when we have a duplication of the input.

In this case, we had a duplication of the input of three and therefore three was paired with more than one output.

Relations and functions can also be represented as sets of points or ordered pairs as we see here.

In example two, we're asked, "Which of the following sets "of ordered pairs represent functions?"

So we could determine whether these are functions or not by analyzing the ordered pairs or if we want, we could put the ordered pairs in a table like Example one.

Remember, for the ordered pairs, the first coordinate, or the x-coordinate, is the input and the second coordinate, or the y-coordinate, is the output.

So for A, notice that zero is paired with negative two, one is paired with four, negative three is paired with three, and five is paired with zero.

Notice how every input is paired with exactly one output.

Therefore A is a function, so we'll say yes.

Let's try one more with a table, then we'll see if you can analyze the ordered pairs to determine if we have a function or not.

So for B, notice that negative four is paired with zero, two is paired with negative three, and two is also paired with negative five.

Notice how this is not a function because the input of two is not paired with exactly one output.

The input of two is paired two outputs, therefore this is not a function, so we say no.

Again, notice how the problem occurred when we had a repetition of the input or the x-coordinate.

So looking at C, let's do this without the table.

So negative five is paired with one, two is paired with one, negative three is paired with one, and zero is paired with one.

Every input is paired with exactly one output and therefore this is a function so the answer is yes.

So it doesn't matter that the inputs are all paired with the same output, every input has exactly one output and therefore it's a function.

Looking at D, notice three is paired with negative four, but then here three is paired with negative two.

So the input of three is not paired with exactly one output, it's actually paired with two outputs and therefore D is not a function, so we can just stop and say no.

For E, we just have one ordered pair and therefore the answer is yes, this is a function.

Every input, there's only one, has exactly one output, in this case, three.

For example three on the graph below, we're asked to plot the points for A, B, C, and D from Example two which I've copied here.

I've also provided the graphs, just to save time.

So I've already plotted the points for A, B, C, and D below.

And we're asked to circle the problem points, which would be the points which make the relation not a function.

So I'm going back to Example two for just a moment.

Notice that B and D were the two that were not functions, so let's focus on those two graphs.

Those two graphs should have problem points.

Looking at the graph of B, notice how these would be the two problem points, because these two points illustrate that the input of two is paired with two outputs.

Two has an output of negative three as well as negative five.

Looking at the ordered pairs above, notice how these are the ordered pairs for the two points that are problem points, where the input of two is paired with two outputs.

Then looking at D, notice how the two problem points would be here because this illustrates that the input of three is paired with two outputs, the output of negative two and the output of negative four.

Looking at D, notice how these two points come from these two ordered pairs where the input of three has an output of negative four and negative two.

So notice how if we had problem points they would be on the same vertical line which brings us to the Vertical Line Test.

The Vertical Line Test tells us, if all vertical lines intersect the graph of a relation in at most one point, that means at zero points or one point, the relation is also a function, meaning one and only one output exists for each input.

If any vertical line intersects the graph of a relation at more than one point, the relation fails the test and is not a function.

More than one value exists for some or all of the input values.

So Example four says, "Use the Vertical Line Test "to determine which of the following graphs are functions."

Looking at our first graph, let's begin sketching vertical lines across the graph.

Very quickly, we can see that these vertical lines intersect the graph in more than one point.

Most vertical lines intersect this graph in two points.

For example, here we have two points of intersection and therefore this fails the Vertical Line Test and is not a function, so we say no.

Looking at the one vertical line, notice how the input value along the horizontal axis has two output values, one here and one here.

This is why this is not a function.

For our second graph, if we begin sketching vertical lines, notice how at no point do these vertical lines intersect the graph in more than one point.

In fact, every time a vertical line intersects this graph in exactly one point, which satisfies the condition of intersecting the graph in at most one point.

Therefore, this is a function so we say yes.

Looking at the next graph, we can quickly see these vertical lines do intersect the graph in more than one point and therefore this is not a function so we say no.

For our last graph, we begin sketching vertical lines.

Notice how all these vertical lines intersect the graph in zero points or one point.

Therefore, this does pass the Vertical Line Test.

All these vertical lines intersect the graph in at most one point, again meaning zero points or one point.

Therefore, this relation is a function, so we say yes.

Now let's talk about the behavior of the graphs of functions.

The graph of a function will always be either increasing, decreasing, or constant.

If a function is increasing, then as the inputs increase, the outputs increase and the graph will be going uphill from left to right.

Here are some examples of some increasing functions.

We can have a line that's going uphill from left to right.

We could also have curves going uphill from left to right.

These are all increasing functions because the graphs are going uphill from left to right.

A decreasing function is a function where when the inputs increase, the outputs decrease and the function would be going downhill from left to right.

So we could have a line going downhill from left to right or we could have curves going downhill from left to right.

Finally, a constant function is where when the inputs change, the outputs remain constant whenever we have a constant function, the graph would be a horizontal line.

Regardless of the input, the output remains the same.

Let's end on defining dependent and independent variables.

In general, we say that the output depends on the input.

Because of this, we call the output variable the dependent variable, and the input variable is called the independent variable, or sometimes the control variable.

If a relation is a function, then we say that the output is a function of the input.

So if we have a function in terms of x and y, where y is the output and x is the input, we could say that y is a function of x.

I hope you found this helpful.