Hello. I'm Professor Von Schmohawk and welcome to Why U.
So far, we have introduced functions which process a single input variable
to produce an output value.
In this lecture we will introduce "multivariable functions"
whose output depends upon multiple input variables.
We saw that with real-valued functions of a real variable
x is typically used as the input variable, also called the "independent variable".
If we call the function "f"
the function's output value is symbolized by the notation "f of x".
When the independent variable is x
the output value is usually assigned to the variable y
called the "dependent variable".
The set of all possible values which can be assigned to the input variable
is called the "domain" of the function
and the set of output values which the function produces
is called the "range" of the function.
For functions whose input and output values can be any real number
the domain and range will both be equal to the set of real numbers, R.
We can think of a function as mapping elements of its domain to elements of its range.
All functions have one domain and one range.
In the Cartesian plane
the domain R corresponds to the 1-dimensional space represented by the x-axis.
Each possible value of the input variable x
corresponds to a specific position on this axis.
Likewise, a range of R corresponds to the 1-dimensional space represented by the y-axis.
And each possible value of the output variable y
corresponds to some position on this axis.
Together, these two positions locate a point in the Cartesian plane
and the collection of all the points in a function's graph
is a curve in this 2-dimensional space.
The curve visually shows how values of the domain are mapped to values of the range.
But functions can also be constructed
whose output depends upon multiple input variables.
In functions of two real variables
x and y are typically chosen as the independent variables
and the output value for a function "f" is written as "f of x and y".
This value is typically assigned to z, the dependent variable.
For functions of a single real variable, the domain is typically R.
But what would be the domain for a function of two real variables?
The values of both variables together determine the function's output value.
But by definition, functions can only map single elements to single elements.
If a function has two input variables
somehow these variables must be combined into a single domain element
which the function can then map to a range element.
This combining of variables
is accomplished by forming the Cartesian product of the two sets of input values.
Forming the Cartesian product of these sets gives us the set "R-two".
R-two contains ordered pairs
whose coordinates are every possible pair of two real numbers.
Each ordered pair is a single input element
which the function can then map to a single output value.
So the domain of a function of two real variables is typically R-two
and for a function which can produce any real value, the range is R.
The domain R-two
corresponds to the 2-dimensional space represented by the xy-plane
and each domain element corresponds to a specific position on this plane.
The range R, corresponds to the 1-dimensional space represented by the z-axis
and each possible value of the output variable z
corresponds to some position on this axis.
Together, these two positions locate a point in 3-dimensional Cartesian space.
And the collection of all the points in a function's graph
forms a surface in this space.
Just as in 2-dimensions
this graph visually shows how values of the domain are mapped to values of the range.
Since the domain is 2-dimensional
and the range is 1-dimensional
this function maps points on a plane to points on a line.
So multivariable functions, like single variable functions
have one domain and one range.
For functions of two real variables, the domain is typically R-two
and if the function can produce any real value, its range is R.
Multivariable functions need not be limited to two input variables.
Functions of three variables are often encountered in Physics and Engineering.
For example, the air temperature at every point in a room
could be represented by a function of three variables.
The three independent variables x, y, and z
could locate a spatial position in the room
and the value of the function could represent the air temperature at that position.
Although it would be difficult to visualize this function using a graph
we can imagine every position in the room containing a number
which represents the temperature at that point in space.
Functions can have any number of input variables
but we will start by examining real-valued functions of a single real variable.
In the next lecture we will introduce "linear functions"
whose graphs appear as lines in the Cartesian plane.