One-to-one functions - Revision history

Lila: /* Resources */

2021-10-25T18:29:43Z

Resources

Lila: /* Resources */

2021-10-25T18:29:19Z

Resources

Lila: /* Resources */

2021-10-05T19:49:13Z

Resources

Lila at 19:47, 5 October 2021

2021-10-05T19:47:06Z

Lila: /* The horizontal line and the algebraic 1-1 test */

2021-10-05T19:46:23Z

The horizontal line and the algebraic 1-1 test

Lila at 19:41, 5 October 2021

2021-10-05T19:41:09Z

Lila at 18:12, 5 October 2021

2021-10-05T18:12:19Z

Rylee.taylor: fixing link to go straight to pdf

2020-08-12T19:00:05Z

fixing link to go straight to pdf

Rylee.taylor at 20:38, 11 August 2020

2020-08-11T20:38:12Z

Rylee.taylor: added one-to-one functions notes by professor Esparza

2020-08-11T20:38:01Z

added one-to-one functions notes by professor Esparza

New page

[https://mathresearch.utsa.edu/wikiFiles/MAT1093/One-to-one%20functions/ One-to-one functions]. Written notes created by Professor Esparza, UTSA.

@@ Line 38: / Line 38: @@
 * [https://www.math.uh.edu/~jiwenhe/Math1432/lectures/lecture01_handout.pdf One-to-one Functions and Inverses], University of Houston
-Licensing
+==Licensing==
 Content obtained and/or adapted from:
 * [https://en.wikibooks.org/wiki/Calculus/Functions Functions, Wikibooks: Calculus] under a CC BY-SA license

← Older revision		Revision as of 18:29, 25 October 2021
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	* [https://en.wikibooks.org/wiki/Calculus/Functions Functions], Wikibooks: Calculus		* [https://en.wikibooks.org/wiki/Calculus/Functions Functions], Wikibooks: Calculus
	* [https://www.math.uh.edu/~jiwenhe/Math1432/lectures/lecture01_handout.pdf One-to-one Functions and Inverses], University of Houston		* [https://www.math.uh.edu/~jiwenhe/Math1432/lectures/lecture01_handout.pdf One-to-one Functions and Inverses], University of Houston
		+
		+	Licensing
		+	Content obtained and/or adapted from:
		+	* [https://en.wikibooks.org/wiki/Calculus/Functions Functions, Wikibooks: Calculus] under a CC BY-SA license

← Older revision		Revision as of 19:49, 5 October 2021
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	==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.
		+	* [https://en.wikibooks.org/wiki/Calculus/Functions Functions], Wikibooks: Calculus
		+	* [https://www.math.uh.edu/~jiwenhe/Math1432/lectures/lecture01_handout.pdf One-to-one Functions and Inverses], University of Houston

@@ Line 8: / Line 8: @@
 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|left|A one-to-one function passes the horizontal line test]]
+[[File:Horizontal-test-ok.png|thumb|A one-to-one function passes the horizontal line test]]
-[[File:Horizontal-test-fail.png|thumb|left|This function does NOT pass the horiontal line test]]
+[[File:Horizontal-test-fail.png|thumb|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:

@@ Line 8: / Line 8: @@
 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]]
+[[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:
@@ Line 32: / Line 32: @@
 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==
 * [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.

@@ Line 1: / Line 1: @@
 To make it simple, for the function <math>f(x)</math>, all of the possible <math>x</math> values constitute the domain, and all of the values <math>f(x)</math> (<math>y</math> on the x-y plane) constitute the range. To put it in more formal terms, a function <math>f</math> is a mapping of some element <math>a\in A</math>, called the domain, to exactly one element <math>b\in B</math>, called the range, such that <math>f:A\to B</math>. The image below should help explain the modern definition of a function:
-[[File:Function_Definition.svg|alt=The image demonstrates a mapping of some element a (the circle) in A, the domain, to exactly one element b in B, the range.|thumb|<math>A</math> is the domain of the function while <math>B</math> is the range. This transformation from set <math>A</math> to <math>B</math> is an example of one-to-one function.]]
 : A function is considered '''one-to-one''' if an element <math>a\in A</math> from domain <math>A</math> of function <math>f</math>, leads to exactly one element <math>b\in B</math> from range <math>B</math> of the function. By definition, since only one element <math>b</math> is mapped by function <math>f</math> from some element <math>a</math>, <math>f:A\to B</math> implies that there exists only one element <math>b</math> 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'''''.
 * [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.

← Older revision		Revision as of 18:12, 5 October 2021
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		+	To make it simple, for the function <math>f(x)</math>, all of the possible <math>x</math> values constitute the domain, and all of the values <math>f(x)</math> (<math>y</math> on the x-y plane) constitute the range. To put it in more formal terms, a function <math>f</math> is a mapping of some element <math>a\in A</math>, called the domain, to exactly one element <math>b\in B</math>, called the range, such that <math>f:A\to B</math>. The image below should help explain the modern definition of a function:
		+	[[File:Function_Definition.svg\|alt=The image demonstrates a mapping of some element a (the circle) in A, the domain, to exactly one element b in B, the range.\|thumb\|<math>A</math> is the domain of the function while <math>B</math> is the range. This transformation from set <math>A</math> to <math>B</math> is an example of one-to-one function.]]
		+	: A function is considered '''one-to-one''' if an element <math>a\in A</math> from domain <math>A</math> of function <math>f</math>, leads to exactly one element <math>b\in B</math> from range <math>B</math> of the function. By definition, since only one element <math>b</math> is mapped by function <math>f</math> from some element <math>a</math>, <math>f:A\to B</math> implies that there exists only one element <math>b</math> 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'''''.
		+
	* [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.

← Older revision		Revision as of 19:00, 12 August 2020
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−	* [https://mathresearch.utsa.edu/wikiFiles/MAT1093/One-to-one%20functions/ 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.

← Older revision		Revision as of 20:38, 11 August 2020
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−	[https://mathresearch.utsa.edu/wikiFiles/MAT1093/One-to-one%20functions/ One-to-one functions]. Written notes created by Professor Esparza, UTSA.	+	* [https://mathresearch.utsa.edu/wikiFiles/MAT1093/One-to-one%20functions/ One-to-one functions]. Written notes created by Professor Esparza, UTSA.