#### 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).

#### 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.

**2) ** must pass the vertical Line Test

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

**3) ** * must pass the Horizontal Line Test

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.

**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.

**Practice** Problems - Part I

- 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.

- 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.

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.

Since the answer is 'yes', this is **not ** a one-to-one function.

Since the answer is 'no', this **is** a one-to-one function.

Since the answer is 'no', this ** is** a 1 to 1 function.

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.

**No**

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