Which of the following sets
of ordered pairs represent a function?
If the set is a function, determine is the function is
increasing, decreasing, constant, or none of these.
Remember for each ordered pair,
the first value is the input, often referred to
as the x-value, and the second value is the output,
often referred to as the y-value.
Let's begin by reviewing the definition of a function
as well as the definitions of increasing, decreasing,
and constant functions.
A function has exactly one output or y-value
for each input or x-value.
And we can use the vertical line test to determine
if a graph represents a function.
And now for the behavior of a function,
a function is increasing if as the inputs
or x-values increase, the outputs or y-values also increase,
which means the graph goes up from left to right.
A function is decreasing if the inputs or x-values increase,
the outputs or y-values decrease
which means the graph goes down from left to right.
And then finally a function is constant
if as the inputs or x-values increase,
the outputs or y-values remain the same or constant.
The graph would be a horizontal line.
Let's go back and take a look at our examples.
The first set was set F that consisted
of five ordered pairs.
Again, notice how set F is given
as five ordered pairs listed horizontally.
Often we're given sets of ordered pairs
in a table or vertically as shown here
which is equivalent.
Notice the ordered pair negative two comma negative four is
here in the table.
The ordered pair negative one comma negative one is
here in the table.
The ordered pair zero comma two is here.
The ordered pair two comma four is here.
And the ordered pair three comma five is here.
Where in the first column we have the inputs
and the second column we have the outputs.
Another way to express a set of ordered pairs would be
to show the mapping which is this representation here.
When we list the inputs here and the outputs here
and then we show the mapping which shows
the input and output relationship.
Where when negative two is the input,
negative four is the output, give by this ordered pair.
When negative one is the input,
negative one is the output given by this ordered pair,
and so on.
The mapping is a nice representation
because now we can easily determine
whether we have a function or not.
Remember, a function has exactly one output
or y-value for each input or x-value.
And notice out here we can see
each input is paired exactly with one output
and therefore we do have a function.
Another way to determine whether we have
a function would be to graph the points given
by the ordered pairs on the coordinate plane
as shown here.
So to determine whether a graph represents a function,
we pass a vertical line across the graph.
If a vertical line ever intersects the graph
in more than one point, we do not have a function.
But notice how if we pass a vertical line
across this graph, we either have one point
of intersection or zero points of intersection,
and therefore this passes the vertical line test
and we do have a function.
And now to describe the behavior of the function.
Notice how from the graph, the graph is going uphill
from left to right which means the function is increasing.
We can also say as the inputs increase,
the outputs also increase.
Let's look at our next example which is set G.
Again we have five ordered pairs listed horizontally
which we can just as easily show in this table here
where we have the inputs in the first column
and the outputs in the second column.
Notice how for this table, the input of one appears twice.
We have an input of one with an output of three
and an input of one with an output of negative three
which means we are not going to have a function here
because remember a function has exactly
one output for each input.
And here we have an input that has two different outputs.
One has an output of three and one has
an output of negative three.
To show the mapping of these ordered pairs,
we would not list one twice in the inputs.
We list it once.
But when showing the mapping, one maps to three
and one also maps to negative three.
And this is the reason why we do not have a function.
Again, the input of one does not have exactly one output.
It has two outputs.
So here we do not have a function.
But look at this graphically and plot
the five ordered pairs as points
on the coordinate plane.
Notice how this graph fails the vertical line test
because at x equals one, this vertical line here,
there is more than one point of intersection.
We have two points of intersection,
and therefore it fails our vertical line test
and we do not have a function.
And because we don't have a function,
we do not describe the behavior.
Let's look at our next example which is set H.
Again here we have set H listed horizontally
and here we have set H listed vertically in a table.
Notice all of the outputs are positive three
so when it comes to the mapping,
we have all the inputs here, we list the output
of three once.
And notice for whatever input we have,
the output is always positive three.
But this still does satisfy the definition
of a function because each input does have
exactly one output.
It's just they happen to share the same output.
But because each input has exactly one output,
this is a function.
And of course we can verify this graphically.
Here are the five ordered pairs graphed
on the coordinate plane.
Using the vertical line test, a vertical line will
never intersect this graph in more than one point,
and therefore it passes the vertical line test
and is a function.
And notice how these five points do line up horizontally
because the output is positive three
and therefore we do have a constant function.
We can also say as the inputs increase,
the outputs remain constant which is why we have
a constant function.
And for the last example, we have set J listed
here horizontally and listed here vertically.
We take a look at the mapping, we have five
different inputs and five different outputs.
Every input has exactly one output
and therefore we do have a function.
We take a look at the five ordered pairs
on the coordinate plane.
Notice how the graph does pass a vertical line test
because a vertical line will never intersect
this graph in more than one point.
And then finally, notice how the graph goes downhill
from left to right, and therefore we have
a decreasing function.
We can also say as the inputs increase,
the outputs decrease.
I hope you found this helpful.