# Analysis of a Function in Graph

## Context
The transcription discusses the concept of functions and how to determine if one variable is a function of another using a specific graph as an example.

## Key Points
1. **Definition of Function**: 
   - A variable y is a function of x if, for every valid input of x, there is exactly one corresponding output of y.
2. **Function Machine Analogy**: 
   - The notion of a 'function machine' is introduced, which illustrates that each input (x) must produce a single output (y).
3. **Graph Example**: 
   - The specific graph under discussion only provides a definition for x when it is -2.
   - Inputting -2 results in multiple y-values: (negative 2, 9), (negative 2, 8), (negative 2, 7), etc.
4. **Conclusion**: 
   - Since the input of -2 yields multiple possible outputs, it is concluded that y is not a function of x in this context.

The transcription concludes that the relationship given in the graph does not satisfy the criteria for a function because multiple outputs are possible for the single input of negative 2.

Khan Academy

In the following graph, is y a function of x?

So, in order for y to be a function of x, for any x that you input into the function, any x for which the function is defined. So let's say we have y is equal to f(x). So we have our little function machine. It should spit out exactly one value of y. If it spits out multiple values of y, we don't know what f(x) is going to be equal to. It could be equal to any of those possible values for y.

So let's see if, for this graph, whether for a given x it spits out exactly one y. Well, the function seems to be only defined, so the domain of this function is x = -2. That's the only place where we have a definition for it. And if we try to input -2 into this little black box, what do we get? Do we get exactly one thing? No. If we put in -2 here, we could get anything. The point (-2, 9) is on this relation. (-2, 8) is on this relation. (-2, 7); (-2, 7.5); (-2, 3.14159)-- they're all on these. So if you put a -2 into this relation, essentially, you actually get an infinite set of values. It could be 9. It could be 3.14. It could be 8. It could be -8. You get an infinite number of results.

So, since it does not map to exactly one output of this function, in the following graph, y is not a function of x.


Does a vertical line represent a function? | Functions and their graphs | Algebra II | Khan Academy

In the following graph,
is y a function of x?

for any x that you input into
the function, any x for which

So let's say we have
y is equal to f of x.

It should spit out
exactly one value of y.

If it spits out multiple values
of y, we don't know what f of x

It could be equal to any of
those possible values for y.

So let's see if, for this
graph, whether for a given x

Well, the function
seems to be only defined

so the domain of this function
is x is equal to negative 2.

That's the only place where
we have a definition for it.

into this little black
box, what do we get?

If we put in negative 2
here, we could get anything.

The point negative 2,
9 is on this relation.

Negative 2, 7; negative 2,
7.5; negative 2, 3.14159--

So if you put a negative 2 into
this relation, essentially,

you actually get an
infinite set of values.

output of this function,
in the following graph,

In the following graph,
is y a function of x?
So in order for y to
be a function of x,
for any x that you input into
the function, any x for which
the function is defined.
So let's say we have
y is equal to f of x.
So we have our little
function machine.
It should spit out
exactly one value of y.
If it spits out multiple values
of y, we don't know what f of x
is going to be equal to.
It could be equal to any of
those possible values for y.
So let's see if, for this
graph, whether for a given x
it spits out exactly one y.
Well, the function
seems to be only defined
so the domain of this function
is x is equal to negative 2.
That's the only place where
we have a definition for it.
And if we try to
input negative 2
into this little black
box, what do we get?
Do we get exactly one thing?
No.
If we put in negative 2
here, we could get anything.
The point negative 2,
9 is on this relation.
Negative 2, 8 is
on this relation.
Negative 2, 7; negative 2,
7.5; negative 2, 3.14159--
they're all on these.
So if you put a negative 2 into
this relation, essentially,
you actually get an
infinite set of values.
It could be 9.
It could be 3.14.
It could be 8.
It could be negative 8.
You get an infinite
number of results.
So since it does not
map to exactly one
output of this function,
in the following graph,
y is not a function of x.

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Does a vertical line represent a function? | Functions and their graphs | Algebra II | Khan Academy

Transcription