Representing Functions and Relations
The given relation is not a function because the x-value 3 corresponds to two y- values. We can also recognize functions as relations where no x-values are. Graphs easily describe the economic relationship between two variables; However, for functional relationships involving three or more variables, you'll find . Describe qualitatively the functional relationship between two quantities by analyzing a graph. Sketch a graph that exhibits the qualitative features of a function.
Let's see what happens when we go over here. If you put 2 into the function, when x is 2, y is negative 2. Once again, when x is 2 the function associates 2 for x, which is a member of the domain. It's defined for 2. It's not defined for 1.
We don't know what our function is equal to at 1. So it's not defined there. So 1 isn't part of the domain. It tells us when x is 2, then y is going to be equal to negative 2. So it maps it or associates it with negative 2. That doesn't seem too troublesome just yet.
Now, let's look over here. Our function is also defined at x is equal to 3. Our function associates or maps 3 to the value y is equal to 2. That seems pretty straightforward. And then we get to x is equal to 4, where it seems like this thing that could be a function is somewhat defined. It does try to associate 4 with things. But what's interesting here is it tries to associate 4 with two different things.
All of a sudden in this thing that we think might have been a function, but it looks like it might not be, we don't know. Do we associate 4 with 5? Or do we associate it with negative 1? So this thing right over here is actually a relation.
You can have one member of the domain being related to multiple members of the range. But if you do have that, then you're not dealing with a function. So once again, because of this, this is not a function.
It's not clear that when you input 4 into it, should you output 5? Or should you output negative 1? And sometimes there's something called the vertical line test that tells you whether something is a function. The parts of a function are called inputs and outputs.
An input is the independent, non-repeating quantity. The output quantity is the dependent quantity.
Recognizing functions from graph (video) | Khan Academy
The value of the output depends on the value of the input. For each input, there is a single output. In the case of tossing a ball in the air, time is the input and height is the output. Remember the last time you were in a parking lot? Is this relation a function?
Can you use the number of cars to correctly figure out the number of tires? Every single car has 4 tires, so the number of tires depends on how many cars are in the parking lot. Every input of cars specifies a single possible output of tires. In this example, the relation of tires to cars is also a function—the number of tires also specifies the number of cars. Now consider a different relation, between houses and the people who live in them.
Describe the functional relationship between two quantities by analyzing a graph
If an address is the input, and the output is the occupants, is this relation also a function? Think of your own house or apartment—are the people staying there always the same? That time you went to camp, the occupancy changed. Every time you had a friend stay over, it changed again. Because a single address can produce more than one set of occupants, the relation is not a function. If you put the input in more than once, are you guaranteed to always get the same output? With the cars and wheels, the answer is yes.
For an input of 25 cars we always get an output of tires, no matter which 25 cars drive into that parking lot or when they arrive. The relation is a function.