Difference between revisions of "One-to-one functions"
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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. | ||
Revision as of 12:12, 5 October 2021
To make it simple, for the function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)} , 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:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A}
is the domain of the function while Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B}
is the range. This transformation from set Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A}
to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B}
is an example of one-to-one 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b} is mapped by function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} from some element Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f:A\to B} implies that there exists only one element Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b} 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.
- One-to-one functions. Written notes created by Professor Esparza, UTSA.
