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.