One to One Function

Range, Domain, horizontal line test

What is a 1 to 1 function?

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 ?

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?

To be a 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"

To be a 1 to 1 function

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 11 ) 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 - Part I

Problem 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) }

If an element in the range repeats, like 14 in function #2, then you do not have a 1 to 1 function

Relation #1 and Relation #3 are both one-to-one functions.

Problem 2

Which functions below are one to one ?

  • Function #1 { (2,1), (4,5), (6,7), (8,9) }
  • Function #2 { (3,6), (8,5), (6,7), (22,6) }
  • Function #3 { (-3,4), (21,-5), (0,0), (8,9) }
  • Function #4 { (9, 19), (34,5), (6,17), (8, 19) }

If an element in the range repeats, like 6 in function #2 or 19 in function #4, then you do not have a 1 to 1 function

Relation #1 and Relation #3 are both one-to-one functions.

Problem 3

Is the function below a one to-one function ?

one to one function

Yes, because every element in the range is matched with only 1 element in the domain.

Practice Problems - Part 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.

Problem 1

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.

Problem 2

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.

Problem 3
One To One Function Problem

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.

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?

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.

Problem 5

Are parabolas 1 to 1?

No

As the picture below shows, parabolas are not one to one

Picture of the horizontal line test
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