Inverse function - Revision history

Sebastian: /* Relevant observations */

2022-04-28T03:59:22Z

Relevant observations

← Older revision		Revision as of 03:59, 28 April 2022
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	== Relevant observations ==		== Relevant observations ==
	* If g is the inverse function of f, then f is the inverse function of g.		* If '''g''' is the inverse function of '''f''', then '''f''' is the inverse function of '''g'''.
	* The domain of f<sup>-1</sup> is the range of f and the range of f<sup>-1</sup> is the domain of f.		* The domain of '''f<sup>-1</sup>''' is the range of '''f''' and the range of '''f<sup>-1</sup>''' is the domain of '''f'''.
	* A function may not have an inverse function, but if it has, the inverse function is unique.		* A function may not have an inverse function, but if it has, the inverse function is unique.

Sebastian: /* Definition */

2022-04-28T03:58:25Z

Definition

← Older revision		Revision as of 03:58, 28 April 2022
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	== Definition ==		== Definition ==
	A function '''g''' is the inverse function of f if '''f(g(x))=x''' for each value of '''x''' in the domain of '''g''', and '''g(f(x))=x''' for each value of '''x''' in the domain of '''f'''. The function '''g''' is denoted as '''f<sup>-1</sup>''' ("inverse of f").		A function '''g''' is the inverse function of f if '''f(g(x))'''='''x''' for each value of '''x''' in the domain of '''g''', and '''g(f(x))=x''' for each value of '''x''' in the domain of '''f'''. The function '''g''' is denoted as '''f<sup>-1</sup>''' ("inverse of f").

	=== Notation ===		=== Notation ===

Sebastian at 03:58, 28 April 2022

2022-04-28T03:58:00Z

← Older revision		Revision as of 03:58, 28 April 2022
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	An inverse function is a function that serves to "undo" another function. That is, if f(x) produces y, then putting y into the inverse of f produces the output x. Not every function has an inverse.		An inverse function is a function that serves to "undo" another function. That is, if '''f(x)''' produces '''y''', then putting '''y''' into the inverse of '''f''' produces the output '''x'''. Not every function has an inverse.

	== Definition ==		== Definition ==

Sebastian: /* Definition */

2022-04-28T03:57:18Z

Definition

← Older revision		Revision as of 03:57, 28 April 2022
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	== Definition ==		== Definition ==
	A function g is the inverse function of f if ~~<math>~~f(g(x))=x~~</math>~~ for each value of x in the domain of g, and g(f(x))=x for each value of x in the domain of f. The function g is denoted as f<sup>-1</sup> ("inverse of f").		A function '''g''' is the inverse function of f if '''f(g(x))=x''' for each value of '''x''' in the domain of '''g''', and '''g(f(x))=x''' for each value of '''x''' in the domain of '''f'''. The function '''g''' is denoted as '''f<sup>-1</sup>''' ("inverse of f").

	=== Notation ===		=== Notation ===
	Whereas the notation used for the inverse function resembles the exponencial notation, the superindex -1 has a distinct use. Therefore, in general, f<sup>-1</sup>(x) ~~is not equal to~~ 1/f(x).		Whereas the notation used for the inverse function resembles the exponencial notation, the superindex -1 has a distinct use. Therefore, in general, '''f<sup>-1</sup>(x)'''≠'''1/f(x)'''.

	== Relevant observations ==		== Relevant observations ==

Sebastian at 03:55, 28 April 2022

2022-04-28T03:55:44Z

← Older revision		Revision as of 03:55, 28 April 2022
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	== Definition ==		== Definition ==
	A function ~~<math>~~g~~</math>~~ is the inverse function of ~~<math>~~f~~</math>~~ if <math>f(g(x))=x</math> for each value of ~~<math>~~x~~</math>~~ in the domain of ~~<math>~~g~~</math>~~, and ~~<math>~~g(f(x))=x~~</math>~~ for each value of ~~<math>~~x~~</math>~~ in the domain of ~~<math>~~f~~</math>~~. The function ~~<math>~~g~~</math>~~ is denoted as ~~<math>~~f~~</math>~~<sup>-1</sup> ("inverse of f").		A function g is the inverse function of f if <math>f(g(x))=x</math> for each value of x in the domain of g, and g(f(x))=x for each value of x in the domain of f. The function g is denoted as f<sup>-1</sup> ("inverse of f").

	=== Notation ===		=== Notation ===
	Whereas the notation used for the inverse function resembles the exponencial notation, the superindex -1 has a distinct use. Therefore, in general, ~~<math>~~f<sup>-1</sup>(x)~~</math>~~ is not equal to 1/f(x).		Whereas the notation used for the inverse function resembles the exponencial notation, the superindex -1 has a distinct use. Therefore, in general, f<sup>-1</sup>(x) is not equal to 1/f(x).

	== Relevant observations ==		== Relevant observations ==

Sebastian: /* Definition */

2022-04-28T03:54:23Z

Definition

← Older revision		Revision as of 03:54, 28 April 2022
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	== Definition ==		== Definition ==
	A function <math>g</math> is the inverse function of <math>f</math> if <math>f(g(x))=x</math> for each value of <math>x</math> in the domain of <math>g</math>, and <math>g(f(x))=x</math> for each value of <math>x</math> in the domain of <math>f</math>. The function <math>g</math> is denoted as f<sup>-1</sup> ("inverse of f").		A function <math>g</math> is the inverse function of <math>f</math> if <math>f(g(x))=x</math> for each value of <math>x</math> in the domain of <math>g</math>, and <math>g(f(x))=x</math> for each value of <math>x</math> in the domain of <math>f</math>. The function <math>g</math> is denoted as <math>f</math><sup>-1</sup> ("inverse of f").

	=== Notation ===		=== Notation ===
	Whereas the notation used for the inverse function resembles the exponencial notation, the superindex -1 has a distinct use. Therefore, in general, <math>f~~</math>~~<sup>-1</sup>(x) is not equal to 1/f(x).		Whereas the notation used for the inverse function resembles the exponencial notation, the superindex -1 has a distinct use. Therefore, in general, <math>f<sup>-1</sup>(x)</math> is not equal to 1/f(x).

	== Relevant observations ==		== Relevant observations ==

Sebastian: /* Definition */

2022-04-28T03:53:40Z

Definition

← Older revision		Revision as of 03:53, 28 April 2022
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	== Definition ==		== Definition ==
	A function g is the inverse function of f if <math>f(g(x))=x</math> for each value of x in the domain of g, and g(f(x))=x for each value of x in the domain of f. The function g is denoted as f<sup>-1</sup> ("inverse of f").		A function <math>g</math> is the inverse function of <math>f</math> if <math>f(g(x))=x</math> for each value of <math>x</math> in the domain of <math>g</math>, and <math>g(f(x))=x</math> for each value of <math>x</math> in the domain of <math>f</math>. The function <math>g</math> is denoted as f<sup>-1</sup> ("inverse of f").

	=== Notation ===		=== Notation ===
	Whereas the notation used for the inverse function resembles the exponencial notation, the superindex -1 has a distinct use. Therefore, in general, f<sup>-1</sup>(x) is not equal to 1/f(x).		Whereas the notation used for the inverse function resembles the exponencial notation, the superindex -1 has a distinct use. Therefore, in general, <math>f</math><sup>-1</sup>(x) is not equal to 1/f(x).

	== Relevant observations ==		== Relevant observations ==

Sebastian: /* Definition */

2022-04-28T03:52:02Z

Definition

← Older revision		Revision as of 03:52, 28 April 2022
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	== Definition ==		== Definition ==
	A function g is the inverse function of f if f(g(x))=x for each value of x in the domain of g, and g(f(x))=x for each value of x in the domain of f. The function g is denoted as f<sup>-1</sup> ("inverse of f").		A function g is the inverse function of f if <math>f(g(x))=x</math> for each value of x in the domain of g, and g(f(x))=x for each value of x in the domain of f. The function g is denoted as f<sup>-1</sup> ("inverse of f").

	=== Notation ===		=== Notation ===

Sebastian: /* Definition */

2022-04-27T23:40:18Z

Definition

← Older revision		Revision as of 23:40, 27 April 2022
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	=== Notation ===		=== Notation ===
	Whereas the notation used for the inverse function resembles the exponencial notation, the superindex -1 has a distinct use. Therefore, in general, f<sup>-1</sup>(x) is not equal to 1/f(x).		Whereas the notation used for the inverse function resembles the exponencial notation, the superindex -1 has a distinct use. Therefore, in general, f<sup>-1</sup>(x) is not equal to 1/f(x).

			== Relevant observations ==
			* If g is the inverse function of f, then f is the inverse function of g.
			* The domain of f<sup>-1</sup> is the range of f and the range of f<sup>-1</sup> is the domain of f.
			* A function may not have an inverse function, but if it has, the inverse function is unique.

Sebastian: /* Definition */

2022-04-27T23:31:08Z

Definition

← Older revision		Revision as of 23:31, 27 April 2022
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	== Definition ==		== Definition ==
	A function g is the inverse function of f if f(g(x))=x for each value of x in the domain of g, and g(f(x))=x for each value of x in the domain of f. The function g is denoted as f<sup>-1</sup> ("inverse of f").		A function g is the inverse function of f if f(g(x))=x for each value of x in the domain of g, and g(f(x))=x for each value of x in the domain of f. The function g is denoted as f<sup>-1</sup> ("inverse of f").

			=== Notation ===
			Whereas the notation used for the inverse function resembles the exponencial notation, the superindex -1 has a distinct use. Therefore, in general, f<sup>-1</sup>(x) is not equal to 1/f(x).