Hello. I'm Professor Von Schmohawk and welcome to Why U.
In the last lecture we introduced functions which process an input variable
which we will call x
to produce an output value.
If we call the function "f"
then the output value it produces is called "f of x".
In algebra, these input and output values are typically real numbers.
In this case, we say that "f is a real-valued function of a real variable".
We saw that the input variable x is called the "independent" variable
since its value does not depend on any other variable.
The independent variable can take on any input value
which the function is capable of processing.
The set of all valid input values is called the "domain" of the function
and the set of output values which the function produces from x
is called the "range" of the function.
If we assign the output value to the variable y
we say that y is the "dependent" variable
since its value depends on the value of the independent variable x.
We can think of the function as "mapping" values in its domain to values in its range.
Although it is possible for a function to map more than one domain value
to the same range value
a function can never map a domain value to multiple range values.
The domain and range of real-valued functions of a real variable
often consist of an infinite number of values.
Since we can't list every value
we represent them using the variables x and y
and visually represent the infinite set of ordered pairs of the function's graph
using the Cartesian plane.
This graph visually shows how each domain value
corresponding to a point on the x-axis
maps to a specific range value
corresponding to a point on the y-axis.
Since each domain value must map to exactly one range value
this limits the shape of the function's graph in certain ways.
For instance, this graph represents a function
since each value of x
maps to a single value of y.
However, this graph does not represent a function
since there are some values of x
which map to two different values of y.
Likewise, this graph does not represent a function
since once again, there are values of x
which map to more than one value of y.
However, if we rotate this graph by ninety degrees then it can represent a function
since any value we pick on the x-axis will map to only one value of y.
There are an infinite variety of real-valued functions of a real variable
and the graphs of these functions can have any imaginable shape
as long as each value in the domain maps to only a single value in the range.
An easy way to test for this condition is the "vertical line test".
Since each x value in a function's domain
must map to no more than one y value in its range
a vertical line drawn at any position on the x axis
must intersect the graph at no more than one point.
To use the vertical line test
take a ruler and move it from one end of the x-axis to the other
while keeping it parallel to the y-axis.
If the ruler intersects the graph more than once for any value of x
the graph does not represent a function.
On the other hand, if the ruler never intersects the graph
more than once for any value of x
it does represent a function.
So this curve represents a function.
However, the same curve rotated by ninety degrees
fails the vertical line test.
Therefore, it does not represent a function.
Any linear graph represents a function
unless the line is vertical.
This curve fails the vertical line test and therefore does not represent a function.
Curves like this which are not functions are still binary relations
since binary relations can have any number of y values
associated with any given value of x.
However, if we eliminate parts of the relation which cause it to fail the test
then its graph can represent a function.
Likewise, this curve does not represent a function
but this curve does.
So far, we have seen functions which process a single input variable
to produce a single output value.
In the next lecture, we will introduce functions which process multiple input variables
from which they produce a single output value.