Difference between revisions of "One-to-one functions"

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Similarly, the horizontal line test, though does not test if an equation is a function, tests if a function is injective (one-to-one). If any horizontal line ever touches the graph at more than one point, then the function is not one-to-one; if the line always touches at most one point on the graph, then the function is one-to-one.
 
Similarly, the horizontal line test, though does not test if an equation is a function, tests if a function is injective (one-to-one). If any horizontal line ever touches the graph at more than one point, then the function is not one-to-one; if the line always touches at most one point on the graph, then the function is one-to-one.
  
[[File:Horizontal-test-ok.png|thumb|A one-to-one function passes the horizontal line test]]
+
[[File:Horizontal-test-ok.png|thumb|left|A one-to-one function passes the horizontal line test]]
  
[[File:Horizontal-test-fail.png|thumb|This function does NOT pass the horiontal line test]]
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[[File:Horizontal-test-fail.png|thumb|left|This function does NOT pass the horiontal line test]]
  
 
The algebraic 1-1 test is the non-geometric way to see if a function is one-to-one. The basic concept is that:
 
The algebraic 1-1 test is the non-geometric way to see if a function is one-to-one. The basic concept is that:
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Using the same method, one can find that <math>a=\pm b</math>, which is not <math>a=b</math>. So, the function is not injective.
 
Using the same method, one can find that <math>a=\pm b</math>, which is not <math>a=b</math>. So, the function is not injective.
 
  
 
==Resources==
 
==Resources==
 
* [https://mathresearch.utsa.edu/wikiFiles/MAT1093/One-to-one%20functions/Esparza%201093%20Notes%201.7.pdf One-to-one functions]. Written notes created by Professor Esparza, UTSA.
 
* [https://mathresearch.utsa.edu/wikiFiles/MAT1093/One-to-one%20functions/Esparza%201093%20Notes%201.7.pdf One-to-one functions]. Written notes created by Professor Esparza, UTSA.

Revision as of 13:46, 5 October 2021

The image demonstrates a mapping of some element a (the circle) in A, the domain, to exactly one element b in B, the range.
is the domain of the function while is the range. This transformation from set to is an example of one-to-one function.

To make it simple, for the function , all of the possible values constitute the domain, and all of the values ( on the x-y plane) constitute the range. To put it in more formal terms, a function is a mapping of some element , called the domain, to exactly one element , called the range, such that . The image below should help explain the modern definition of a function:

A function is considered one-to-one if an element from domain of function , leads to exactly one element from range of the function. By definition, since only one element is mapped by function from some element , implies that there exists only one element from the mapping. Therefore, there exists a one-to-one function because it complies with the definition of a function. This definition is similar to Figure 1.

The horizontal line and the algebraic 1-1 test

Similarly, the horizontal line test, though does not test if an equation is a function, tests if a function is injective (one-to-one). If any horizontal line ever touches the graph at more than one point, then the function is not one-to-one; if the line always touches at most one point on the graph, then the function is one-to-one.

A one-to-one function passes the horizontal line test
This function does NOT pass the horiontal line test

The algebraic 1-1 test is the non-geometric way to see if a function is one-to-one. The basic concept is that:

Assume there is a function . If:

, and , then

function is one-to-one.

Here is an example: prove that is injective.

Since the notation is the notation for a function, the equation is a function. So we only need to prove that it is injective. Let and be the inputs of the function and that . Thus,


So, the result is , proving that the function is injective.

Another example is proving that is not injective.

Using the same method, one can find that , which is not . So, the function is not injective.

Resources