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In this lesson we will discuss one-to-one functions. Before covering what
a one-to-one function is, I will first review the definition of a function.
Function:
- a connection between sets in which each element of the first set
corresponds with exactly one element of the second set (each input
produces exactly one output)
o an example of a function is the set of students at Purdue and
their student identification numbers
๏ง each student has exactly one student identification number
and each student identification number corresponds with
exactly one student (one-to-one function)
- the graph of a function must pass the vertical line test
o a vertical line (which represents an input) can only intersect the
graph of a function once
An example of a function is ๐
2
. Remember that to find a function
value we simply replace ๐ฅ with a number or an expression. To find ๐
we replace ๐ฅ with 2.
2
To find ๐
, we replace ๐ฅ with โ 2.
2
Notice that for the function ๐, both inputs ( 2 and โ 2 ) produce the same
output (4); ๐ is a function because each input results in exactly one output,
however it is not a one-to-one function because each output is not the
result of exactly one input. The output 4 is the result of two different
inputs 2 and โ 2.
One-to-one function:
- a function in which each output is the result of exactly one input
o every input has exactly one output (this makes it a function) and
every output is the result of exactly one input (this makes it one-
to-one)
Showing algebraically that one output is the result of more than one input
as I did earlier with ๐
= 4 and ๐
= 4 is one way to determine
whether a function is one-to-one or not. Another option to determine
whether a function is one-to-one or not is to use its graph; for that, we use
the horizontal line test.
Horizontal line test :
- a horizontal line (which represents an output) can only intersect the
graph of a function once in order for that function to be one-to-one
The vertical line test is used to determine whether a graph represents a
function, and the horizontal line test is used to determine whether a
function is one-to-one or not.
Example 1 : Given the following graph, determine whether this is the
graph of function. If it is, then determine whether this is the graph of a
one-to-one function.
Keep in mind that each of these
problems in LON-CAPA will always
ask if the graph is a function first,
before asking if itโs a one-to-one
function. So you should never answer
No then Yes, because that would mean
that you have a graph that is not a
function, but is somehow a one-to-one
function. Anytime you answer No to
the first part, you should always
answer no to the second part as well.
A graph must first be a function,
before it can be a one-to-one function.
Example 4 : Given the following graph, determine whether this is the
graph of function. If it is, determine whether it is a one-to-one function.
Example 5 : Given the following graph, determine whether this is the
graph of function. If it is, determine whether it is a one-to-one function.
YES, this is the graph of a
function
NO, this is not the graph of a
one-to-one function
As shown on the previous
example, a parabola passes the
vertical line test, but not the
horizontal line test. In order to
make a quadratic function one-
to-one, its domain must be
restricted.
YES, this is the graph of
a function
YES, this is the graph of
a one-to-one function
Since the domain of this
quadratic function has
been restricted, this graph
both passes the vertical
line test and the horizontal
line test. Restricting the
domain of this function is
what limits the graph to
essentially half a parabola.
Example 6 : Given the following graph, determine whether this is the
graph of function. If it is, determine whether it is a one-to-one function.
Example 7 : Given the following graph, determine whether this is the
graph of function. If it is, determine whether it is a one-to-one function.
Once again, you should
never answer No then Yes
on any of these problems (in
the notes or on the
homework), because that
would mean that you have a
graph that is not a function,
but is somehow a one-to-one
function. Anytime you
answer No to the first part,
you should always answer
no to the second part as
well. A graph must first be
a function, before it can be a
one-to-one function.
Another tool for determining what is (what is not) a one-to-one function is
the theorem on increasing/decreasing functions.
Theorem on Increasing/Decreasing Functions :
- a function that is strictly increasing or strictly decreasing throughout
its domain is one-to-one (no turning points)
o all linear functions are one-to-one because they are either always
increasing or always decreasing
o the graph of a quadratic function is parabola which is both
increasing and decreasing, so it is not one-to-one
๏ง as shown before, the way we make it a one-to-one function
is by restricting its domain so itโs either always increasing
or always decreasing
Basically as long as the graph of a function has no turning points, it is the
graph of a one-to-one function.
In the next set of notes weโll be covering Inverse Functions, and how to
find an inverse. Keep in mind that ONLY one-to-one function have an
inverse, so that is why it is important to understand and identify one-to-
one functions.
Neither of these parabolas represent the graphs of one-to-one functions
because neither of these graphs is strictly increasing or strictly
decreasing (both have turning points).
