Hello. I'm Professor Von Schmohawk and welcome to Why U.
In the last lecture we introduced a special type of binary relation called a function.
We saw that functions associate elements from a set of inputs called the "domain"
to elements of a set of outputs called the "range".
The elements of the domain and range can be any type of objects.
We saw that functions can be described by a set of ordered pairs called the "graph"
where the first element of each ordered pair is a domain element
and the second element is the range element to which it is related.
Typically in algebra, a function’s domain and range are sets of real numbers
and the function's graph is a set of ordered pairs of real numbers.
This allows us to visually represent the function's graph as points on the Cartesian plane.
This visual representation of ordered pairs
is also referred to as the "graph" of the function.
Defining a function by listing all the ordered pairs in its graph
is only practical if there are a reasonably small number of ordered pairs.
So how do we define functions which consist of an infinite number of ordered pairs?
Since the line in this graph contains an infinite number of points
it represents an infinite number of ordered pairs.
Functions like this obviously cannot be described by listing every ordered pair in the graph.
Instead, we can typically state a condition which every ordered pair must satisfy.
This condition can usually be stated mathematically as an equation.
Let’s say that we wish to define the function represented by this graph.
Let’s call the coordinates of a typical ordered pair, x and y.
Notice that for every point on this line
the y-coordinate has a value twice that of the x-coordinate.
For instance, if we pick an x value of two
the y value will be four.
Likewise, if we pick an x value of three
y will be six.
Or if x is negative two
y will be negative four.
So the statement that y must equal two times x
is a condition which describes the function illustrated by this graph.
If we like, we can define this function formally using the language of set notation.
Since a function is a set of ordered pairs
the formal definition is a description of this set.
We can define this set of ordered pairs as follows:
Function "f"
is the set of all ordered pairs with elements x and y
such that x is a real number
and y is equal to two x.
In this definition, the only limitation placed on x is that it must be a real number.
Therefore, the first coordinate of an ordered pair can have any real value.
In addition, this definition says that y must have a value twice that of x.
So the second coordinate of each ordered pair must have a value of twice the first coordinate.
Together, these requirements limit the ordered pairs in this set
to any real pair whose second coordinate is twice its first coordinate.
And the graphical representation of this set is the graph of the function.
In the last lecture we said that a function can be thought of as a machine.
Let’s call this function "f".
When we give this machine an input value symbolized by x
it produces an output value symbolized by "f of x".
In the function which we just discussed
the output "f of x" is always equal to two times the input, x.
Therefore, the statement "f of x equals two x"
is another way to describe the function "f".
This statement says that "f" is a function whose output value is always twice its input value.
Although this "function notation" is a common way to describe numeric functions
the formal definition of this function using set notation is a more complete description.
It not only defines how elements x are mapped to elements y
but it also tells us that the domain of the function is the set of real numbers R
since the input values x must be members of R.
Mathematicians agree that any rigorous definition of a function is incomplete
without stating the domain of the function.
For example, changing the domain to Z, the set of integers
creates a different function.
However, in algebra the domain is usually assumed to be the set of real numbers
unless otherwise stated.
Therefore, when the domain of a function is R
this part of the definition is often left out
or more commonly, functions are simply described by an equation.
In this equation, the symbols x and y are called "variables".
A variable is a symbol which can take on any value within some set of possible values.
Variables whose values can be any real number are called "real variables".
In algebra texts, variables are often assumed to be real unless otherwise stated.
In the function we discussed
the value of y
is determined by the value of x.
Since y’s value is dependent on x
y is called the "dependent variable"
and x is called the "independent variable".
In this function the independent variable x
can be any real number.
This equation tells us that y must always be twice the value of x.
Therefore, if x can be any real number, so can y.
Since the input x and the output y can be any real number
both the domain and range of this function are the set of real numbers, R.
In numeric functions the codomain is typically chosen
to be the set containing the general type of number which makes up the range.
Therefore, we will also define the codomain to be R
the same as the range.
We write the notation "f colon R arrow R"
to indicate a mapping by function "f" from the domain to the codomain.
This notation tell us that both the domain and codomain of f are the set of real numbers.
This is read as "f is a function from R to R"
or "f is a function on R ".
Or we can say that "f is a real-valued function of a real variable".
When the domain of a function is the entire set of real numbers
every position on the x-axis will be associated with some value of y
which determines the height of the graph above or below that position.
We have seen that not every binary relation is a function.
Although binary relations can associate a single domain element
to multiple elements in the codomain
a function cannot.
In the next lecture we will introduce a simple graphical test
to see whether a binary relation is a function.