Hey everyone! It's Justin again, and welcome back to our functions unit.
In the last lesson, we discovered that functions are a relationship between inputs and outputs. The
thing is, functions aren't just any relationship. They are a specific relationship where each input
is related to exactly one output. We're gonna unpack that with a practical example before
we dive into recognizing if a relationship is indeed a function or not. Let's get into it!
[Music]
Imagine you're at the grocery store. You have a few items to buy: a loaf of bread, a jar of
peanut butter, some jelly, a bag of lemons, and laundry detergent. You head to the self-checkout
line and begin to scan your items. The bread comes up as $5.99, peanut butter as $3.99,
jelly as $6.59, lemons also as $3.99, and finally, the laundry detergent as $14.49.
But hold on a second. Is it all right that the peanut butter and the lemons
have the same price? Yeah, for sure. I mean, there are probably many different
items at the grocery store that will have the same price. In this scenario,
our inputs have been mapped to our outputs. This forms a function because each item has one price.
So each input is related to exactly one output, even if some of those outputs are the same.
But what if while you're checking out your items, you decide to grab a second bottle of the exact
same laundry detergent and you scan it, expecting $14.49 again, but instead it's $25.49! Well,
that doesn't seem right! In the same way, a single input corresponding to multiple
outputs doesn't fit the definition of a function. It's not just impractical at the grocery store;
it's mathematically inconsistent. In math, like at the grocery store,
we expect consistency. One input gives one output. If this isn't the case, it is not a function.
So you can think about functions kind of like a grocery checkout system where each item has
a single price. Translated into math, each input is related to exactly one output. Now
take a moment to practice this concept with the grocery checkout activity. Pause here,
give it a go, and we'll review the answers together back here when you're ready.
Here's the first relation from the grocery checkout activity. This relation is a function
as each input corresponds to exactly one output. Having different inputs share the
same output is totally fine because different grocery items, like the carrots and broccoli,
are allowed to have the same price. It's entirely possible and probably quite common at the store.
The second relation is not a function as the same input is linked to two different outputs.
Two of the same items at the grocery, like these two identical bottles of orange juice,
cannot ring up at two different prices. It's just not possible.
And lastly, the third relation is a function because each input links to exactly one output.
Two of the same items at the grocery store, like the two identical packets of pasta,
should most definitely ring up at the same price. Nice work!
In math, we can depict the relations between inputs and
outputs in a variety of ways. One method, as shown in our grocery checkout example,
is called a mapping diagram. But we could also use a table like we saw in our last
lesson. Pause the video and take a moment to determine if this relation is a function or not.
This is a function since each input links to one output,
and it's totally all right that all three inputs map to the same output. That would
be like three items at the grocery store having the same price -- totally possible.
A second way to represent relations is as a list of coordinates, like this. Coordinates are written
in the form x comma y, where the x coordinate is the input, and the y coordinate is the output. You
can analyze these coordinates directly or convert them into an input-output table like this if it
makes it easier for you to analyze. Similarly here, the x coordinate is the input, and the
y coordinate is the output. Pause the video here and analyze if this relation is a function or not.
Regardless of which model you examine, this relation is not a function,
and that's because each input does not relate to exactly one output! In fact,
this one input of 2 relates to three different outputs: 5, 3, and 1. And we
know by now that identical items at the grocery store cannot ring up at three different prices.
Let's keep working with this relation that we now know is
not a function. Since a relation can be represented as coordinates,
well, then that means that a relation can also be represented on a coordinate plane
as a graph. Let's plot these coordinate points on our graph and see what happens.
Remember the x coordinates are our inputs and the y coordinates are our outputs. So starting with
our first coordinate of (2, 5), if we input 2 on our x axis, our output is 5 on our y axis, which
means our point is right here. For our second coordinate of (3, 7), if we input 3 on our x axis,
our output is 7 on our y axis, which means our point is right here. Let's do one more together,
(4, 9). So we input 4 and we get an output of 9, giving us a point right here. Great!
I'm gonna go ahead and plot those last four points for us. There they are. All of these
coordinates have been plotted on our graph, and we can see the visual representation of our relation.
We previously established that this is not a function since
the single input of 2 corresponds to three different outputs, 5, 3,
and 1. So these are the points that make this relation not a function.
Do you have a prediction about how we can tell if a relation is a function or not from
its graph? Let's slide this over and look at another one to see if your prediction's right.
Relation 6 is also not a function. I'd like you to pause the video here and take a look
at Relation 6 and see if you can analyze why this relation is also not a function.
Well, similar to the graph of Relation 5 on the left, Relation 6 also has one input that
corresponds to more than one output. Can you see where that is happening?
These are my problematic coordinates and that’s because I have one input of 3 here,
that gives me not one, not two, not even three, but four different outputs!
These two graphs both show relations that are not functions, because of these… let’s
call them problem areas. Pause the video here and think back to your
prediction. What do the problem areas on both of these graphs have in common?
Well, in both relations, there was a single input on the x axis
that corresponded to multiple outputs on the y axis, basically, points are stacked vertically,
one above or below the other. So we can imagine drawing vertical lines on our graph and if
one of those vertical lines intersects more than one point, like they do here,
we know there's more than one output for an input, meaning it's not a function.
This method is known as the vertical line test. It's a visual test used to check if a
graphed relation is a function. Basically, if any vertical line intersects the graph more than once,
then the relation isn't a function. Let's practice this test with more graph relations!
In these example problems, we'll want to use the vertical line test to determine if the relations
graphs below are functions or not. Now, as you can see, these relations are represented by
lines and curves, not just a few points like our previous examples. And that's totally OK because,
remember, lines -- just like curves -- are made up of infinitely many points squished
together. So these graphs are essentially just showing a vast number of points.
OK, let's start by using the vertical line test on Relation 7. We can either
imagine drawing vertical lines across our graph, but we can physically draw them
too. I like to visually scan my graph from left to right with my vertical lines like
this. And everything looks good. I'm only seeing one intersection at a time. Cool!
After scanning my graph with the vertical line test, I did not find any place where my vertical
line intersected with my graph more than once, which means that this relation is in fact a
function. Pause the video here and give relations 8, 9, and 10 a try on your own in your PDF.
In Relation 8, did you find any vertical line test that failed?
Me too. Here's one place where I found it failed, because the graph of my relation
crossed my vertical at two points here and here. That means this relation is not a
function! What about Relation 9? Did you find any vertical line test that failed?
Yep, same here. This is one of many portions of the graph that failed since it intersects
my vertical line at two points. So since there is one input that has two outputs,
this relation is also not a function.
And finally, Relation 10. What did you find? Hmm, well, when I did my vertical
line test scanning across my graph, I didn't find any part where there was more than one
output for an input. So this relation, Relation 10, is a function. Great work!
In today's lesson, you learn how to determine if a relation is a function,
given the variety of ways a relation could be presented: as mapping diagrams, tables,
sets of coordinates, or graphs. In each case, no matter how the relation is presented,
we always want to inspect that each input has exactly one output. And unfortunately,
if that is not the case, then, well, it's not a function!
Once you're all done watching this video and completing the guided notes,
check out the additional practice in your PDF. It'll prepare you for
the practice game that follows. And that's all for me today. I'll see you next time!