- What are injection functions for?
- Function conditioning
- Examples of injection functions with solved exercises
- Example 1
- Example 2
- Example 3
- Example 4
- Example 5
- Example 6
- References
An injective function is any relation of elements of the domain with a single element of the codomain. Also known as a one-to-one function (1 - 1), they are part of the classification of functions with respect to the way their elements are related.
An element of the codomain can only be the image of a single element of the domain, in this way the values of the dependent variable cannot be repeated.
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A clear example would be grouping men with jobs in group A, and in group B all the bosses. Function F will be the one that associates each worker with his boss. If each worker is associated with a different boss through F, then F will be an injective function.
To consider a function injective, the following must be met:
∀ x 1 ≠ x 2 ⇒ F (x 1) ≠ F (x 2)
This is the algebraic way of saying For every x 1 different from x 2 we have an F (x 1) different from F (x 2).
What are injection functions for?
Injectivity is a property of continuous functions, since they ensure the assignment of images for each element of the domain, an essential aspect in the continuity of a function.
When drawing a line parallel to the X axis on the graph of an injective function, the graph should only be touched at a single point, no matter at what height or magnitude of Y the line is drawn. This is the graphical way to test the injectivity of a function.
Another way to test if a function is injective is by solving the independent variable X in terms of the dependent variable Y. Then it must be verified if the domain of this new expression contains the real numbers, at the same time as for each value of Y there is a single value of X.
The functions or order relations obey, among other ways, the notation F: D f → C f
What is read F that goes from D f to C f
Where the function F relates the sets Domain and Codomain. Also known as the starting set and the finishing set.
The domain D f contains the allowed values for the independent variable. The codomain C f is made up of all the values available to the dependent variable. The elements of C f related to D f are known as the Range of the function (R f).
Function conditioning
Sometimes a function that is not injective can be subjected to certain conditions. These new conditions can make it an injective function. All kinds of modifications to the domain and codomain of the function are valid, where the objective is to fulfill the injectivity properties in the corresponding relationship.
Examples of injection functions with solved exercises
Example 1
Let the function F: R → R be defined by the line F (x) = 2x - 3
A:
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It is observed that for every value of the domain there is an image in the codomain. This image is unique which makes F an injective function. This applies to all linear functions (Functions whose highest degree of the variable is one).
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Example 2
Let the function F: R → R be defined by F (x) = x 2 +1
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When drawing a horizontal line, it is observed that the graph is found on more than one occasion. Due to this the function F is not injective as long as R → R is defined
We proceed to condition the domain of the function:
F: R + U {0} → R
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Now the independent variable does not take negative values, in this way repeating results is avoided and the function F: R + U {0} → R defined by F (x) = x 2 + 1 is injective.
Another homologous solution would be to limit the domain to the left, that is, to restrict the function to only take negative and zero values.
We proceed to condition the domain of the function
F: R - U {0} → R
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Now the independent variable does not take negative values, in this way repeating results is avoided and the function F: R - U {0} → R defined by F (x) = x 2 + 1 is injective.
Trigonometric functions have wave-like behaviors, where it is very common to find repetitions of values in the dependent variable. Through specific conditioning, based on prior knowledge of these functions, we can narrow the domain to meet the conditions of injectivity.
Example 3
Let the function F: → R be defined by F (x) = Cos (x)
In the interval the cosine function varies its results between zero and one.
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As can be seen in the graph. It starts from zero at x = - π / 2, then reaches a maximum at zero. It is after x = 0 that the values begin to repeat, until they return to zero at x = π / 2. In this way it is known that F (x) = Cos (x) is not injective for the interval.
When studying the graph of the function F (x) = Cos (x), intervals are observed where the behavior of the curve adapts to the injectivity criteria. Such as the interval
Where the function varies results from 1 to -1, without repeating any value in the dependent variable.
In this way the function function F: → R defined by F (x) = Cos (x). It is injective
There are nonlinear functions where similar cases occur. For expressions of rational type, where the denominator contains at least one variable, there are restrictions that prevent the injectivity of the relationship.
Example 4
Let the function F: R → R be defined by F (x) = 10 / x
The function is defined for all real numbers except {0} who has an indeterminacy (It cannot be divided by zero) .
As the dependent variable approaches zero from the left it takes very large negative values, and immediately after zero, the values of the dependent variable take large positive figures.
This disruption makes the expression F: R → R defined by F (x) = 10 / x
Don't be injective.
As seen in the previous examples, the exclusion of values in the domain serves to "repair" these indeterminacies. We proceed to exclude zero from the domain, leaving the starting and finishing sets defined as follows:
R - {0} → R
Where R - {0} symbolizes the reals except for a set whose only element is zero.
In this way the expression F: R - {0} → R defined by F (x) = 10 / x is injective.
Example 5
Let the function F: → R be defined by F (x) = Sen (x)
In the interval the sine function varies its results between zero and one.
Source: Author.
As can be seen in the graph. It starts from zero at x = 0 and then reaches a maximum at x = π / 2. It is after x = π / 2 that the values begin to repeat, until they return to zero at x = π. In this way it is known that F (x) = Sen (x) is not injective for the interval.
When studying the graph of the function F (x) = Sen (x), intervals are observed where the behavior of the curve adapts to the injectivity criteria. Such as the interval
Where the function varies results from 1 to -1, without repeating any value in the dependent variable.
In this way the function F: → R defined by F (x) = Sen (x). It is injective
Example 6
Check if the function F: → R defined by F (x) = Tan (x)
F: → R defined by F (x) = Cos (x + 1)
F: R → R defined by the line F (x) = 7x + 2
References
- Introduction to Logic and Critical Thinking. Merrilee H. Salmon. University of Pittsburgh
- Problems in Mathematical Analysis. Piotr Biler, Alfred Witkowski. University of Wroclaw. Poland.
- Elements of Abstract Analysis. Mícheál O'Searcoid PhD. Department of mathematics. University college Dublin, Beldfield, Dublind 4.
- Introduction to Logic and to the Methodology of the Deductive Sciences. Alfred Tarski, New York Oxford. Oxford University press.
- Principles of mathematical analysis. Enrique Linés Escardó. Editorial Reverté S. A 1991. Barcelona Spain.