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One to One Function

Range, Domain, horizontal line test

What is a 1 to 1 function?

question
Look at the two functions below, ( # 1 and #2). They only differ by a single number.
One of the functions is a one to one function, and the other is not.
Which function, do you think, is 1 to 1 ?
question
A one to one function is a function in which every element in the range corresponds with one and only one element in the domain.
So, #1 is not one to one because the range element 5 goes with 2 different values in the range (4 and 11) .
example of one to graph, chart, example

How to know if f(x) is 1 to 1?

question
To be a function
To be a 1 to 1 function
1) each element in domain must go to a unique range element
In our prior lesson, we said that the domain elements cannot repeat.

relation vs function set of points




2) must pass the vertical Line Test

vetical line test picture
As we learned in our vertical line test lesson, this is really the exact same as saying "elements in the domain cannot repeat"
1) must satisfy requirements for function

2) * each element in range must go to a unique element in the domain
One to One function points



3) * must pass the Horizontal Line Test

Picture of horizontal line test graph

The horizontal line test is really just a re-statement of statement 2 above that
" each element in range must go to a unique element in the domain "
Arrow Chart of 1 to 1 vs regular Function
Below you can see an arrow chart diagram that illustrates the difference between a regular function and a one to one function. Function #1 is not a 1 to 1 because the range element of '5' goes with two different elements( 4 and 5 ) in the domain.
one to one function arrow chart example
Graph Example
We can also look at the graphs of functions and use the horizontal line test to determine that the given function is not 1 to 1.
Picture of the horizontal line test
Practice Problems I


1) Which functions below are 1 to 1?
    Function #1 { (2,27), (3,28), (4,29), (5,30) }
    Function #2 { (11,14), (12,14) , (16,7), (18,13) }
    Function #3 { (3,12), (4,13), (6,14), (8,1) }
Answer

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Practice Problems II


Use the horizontal line test and your knowledge of 1 to 1 functions to determine whether or not each graph below is 1 to 1 .
1)
Answer
Ask yourself:
"Can I draw a horizontal line (anywhere) that will hit the graph two times ?"
Since the answer is 'yes', this is not a one-to-one function.


2)
Answer
Ask yourself:
"Can I draw a horizontal line (anywhere) that will hit the graph two times ?"
Since the answer is 'no', this is a one-to-one function.


3)
Answer
Ask yourself:
"Can I draw a horizontal line (anywhere) that will hit the graph two times ?"
Since the answer is 'no', this is a 1 to 1 function.
One To One Function Problem

4) Are all lines one to one like the prior problem was? If not, which types of lines are one to one and which types are not?
Answer

Vertical lines such as x= 2 are not functions at all.
Horizontal lines such as y = 9 are functions but they are not 1 to 1 functions.
All other lines are indeed one to one functions


5) Are parabolas 1 to 1?
Answer
No
As the picture below shows, parabolas are not one to one
Picture of the horizontal line test
Further Reading