- [Voiceover] We want to find the domain and range
of the relation given here as a set of ordered pairs.
And then we're asked to determine
whether the relation is a function.
When we have a relation given as a set of ordered pairs,
each ordered pair represents and input,
and the corresponding output.
And therefore, because the domain
is a set of all possible inputs,
the domain of the given relation
will be the set containing
all of the first values in the ordered pairs.
So the domain would be the set containing
five, four, three, and two.
When we give the domain, though, we use these braces
and we should list the values in order
from least to greatest.
So let's say the domain is the set containing
two, comma, three, comma, four, comma, five.
And because the range is a set of all possible outputs,
the range would be the set containing
all of the second values of the ordered pairs.
So the range would be the set containing
negative seven, and from here notice how
the output values are all negative seven,
and therefore the range is the set containing
the one value of negative seven.
Notice how we don't list negative seven four times.
And now we'll test to see if the relation is a function.
A function is a special relation,
where a function is a relation in which
each input has exactly one output.
To help us determine whether this relation
is a function, let's show the relation using a mapping,
where this set will be the domain of the set
of all possible inputs.
And this will be the range of the set
of all possible outputs.
So I'll list the values or elements of
two, three, four, and five in this set,
and the value of negative seven in this set.
Now if we go to the pair five, comma, negative seven,
this tells us when the input is five,
the output is negative seven,
and therefore we draw an arrow from five
to negative seven.
When the input is five, the corresponding
output is negative seven.
For the ordered pair four, comma, negative seven,
when the input is four, the output is negative seven
so we draw an arrow from four to negative seven,
for the ordered pair three, comma, negative seven
we draw an arrow from three to negative seven,
and for the ordered pair two, comma, negative seven
we draw an arrow from two to negative seven.
Notice how it's very clear now that each input
has exactly one output.
It is true that all the inputs have the same output,
but each input does have exactly one output,
and therefore the relation is a function.
Notice how, if the relation was not a function,
at least one of the inputs would have to have
two arrows pointing to two different outputs.
The second way to test to see if a relation is a function
is to look at the graph of the relation.
So because we have four ordered pairs,
we would have four points on the
coordinate plane as we see here,
where the inputs are along the horizontal axis,
often the x axis,
and the outputs are along the vertical axis,
often the y axis.
Graphically we use what's called a vertical line test
to determine whether a relation is a function.
The vertical line test states
that if a vertical line intersects a graph
of a relation more than once,
then the relation is not a function.
If you were to sketch vertical lines across the graph,
notice how they would never intersect the graph
in more than one point.
They would either intersect the graph
in zero points or one point,
but the vertical line passes through one of the four points.
So because that vertical line will never intersect
this graph more than once,
this passes the vertical line test,
and the relation is a function.
So we could say the graph passes
the vertical line test, the relation is a function.
I hope you found this helpful.