Difference between revisions of "Function"
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− | A function is a rule that maps one set of values to another set of values. For instance one function may map 1 to 1, 2 to 4, 3 to 9, 4 to 16, and so on. This function has the rule that it takes its input value, and squares it to get an output value. Lets call this function <math>f</math>. A common notation to define <math>f</math> is: <math>f(x) = x^2</math>. This tells us that <math>f</math> is a function that squares | + | A function is a rule that maps one set of values to another set of values. For instance, one function may map 1 to 1, 2 to 4, 3 to 9, 4 to 16, and so on. This function has the rule that it takes its input value, and squares it to get an output value. Lets call this function <math>f</math>. A common notation to define <math>f</math> is: <math>f(x) = x^2</math>. This tells us that <math>f</math> is a function that squares its argument (its input value). Note that this "rule" can be arbitrarily complicated and doesn't need to be given by a simple formula or description. The only requirement is that <math>f(x)</math> should be uniquely determined by <math>x</math>. The following are examples of functions: |
<math>\displaystyle f(x)=x ^ {2}+2x-2</math> | <math>\displaystyle f(x)=x ^ {2}+2x-2</math> | ||
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===Domain and Range=== | ===Domain and Range=== | ||
− | The domain of a function is the [[set]] of input values for the argument of a function. The range of a function is the [[set]] of output values for that function. For an example, consider the function <math>f(x) = \sqrt{x^2-9}</math>. The domain of the function is the set <math>{x:|x|>3}</math> where <math>x</math> is a real number. The range is the set of all non-negative real numbers | + | The domain of a function is the [[set]] of input values for the argument of a function. The range of a function is the [[set]] of output values for that function. For an example, consider the function <math>f(x) = \sqrt{x^2-9}</math>. The domain of the function is the set <math>{x:|x|>3}</math>, where <math>x</math> is a real number. The range is the set of all non-negative real numbers because the square root can never return a negative value. |
===Real functions=== | ===Real functions=== | ||
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===The graph of a function=== | ===The graph of a function=== | ||
− | Functions are often graphed To find out if a graph is a function, it must stand up to the [[vertical line test]]. | + | Functions are often graphed. To find out if a graph is a function, it must stand up to the [[vertical line test]]. |
===The Inverse of a Function=== | ===The Inverse of a Function=== | ||
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===Injections, surjections, bijections=== | ===Injections, surjections, bijections=== | ||
− | *An [[injective function]](or one-to-one) is a function which has distinct values for distinct arguments. | + | *An [[injective function]] (or one-to-one) is a function which has distinct values for distinct arguments. |
− | By definition, <math>\displaystyle f:A\to B</math> is injective if <math>\displaystyle f(a)=f(b) \Rightarrow a=b </math>, or equivalently, <math>\displaystyle a\neq b \Rightarrow f(a)\neq f(b) </math> | + | By definition, <math>\displaystyle f:A\to B</math> is injective if <math>\displaystyle f(a)=f(b) \Rightarrow a=b </math>, or equivalently, <math>\displaystyle a\neq b \Rightarrow f(a)\neq f(b) </math>. |
− | If <math>\displaystyle A</math> and <math>\displaystyle B</math> are finite sets injectivity implies <math>\displaystyle |A|\leq |B|</math>. | + | If <math>\displaystyle A</math> and <math>\displaystyle B</math> are finite sets, injectivity implies <math>\displaystyle |A|\leq |B|</math>. |
===Monotonic functions=== | ===Monotonic functions=== | ||
− | A function <math> \displaystyle f:A\to B</math> is called [[monotonically increasing]] if <math>\displaystyle f(x_1)\geq f(x_2) </math> holds whenever <math>\displaystyle x_1>x_2 </math>. If the inequality holds strictly <math>\displaystyle (f(x_1)>f(x_2)) </math> | + | A function <math> \displaystyle f:A\to B</math> is called [[monotonically increasing]] if <math>\displaystyle f(x_1)\geq f(x_2) </math> holds whenever <math>\displaystyle x_1>x_2 </math>. If the inequality holds strictly <math>\displaystyle (f(x_1)>f(x_2)) </math>, |
then the function is called [[strictly increasing]]. | then the function is called [[strictly increasing]]. | ||
− | Similarlly, a function <math> \displaystyle f:A\to B</math> is called [[monotonically decreasing]] if <math>\displaystyle f(x_1)\geq f(x_2) </math> holds whenever <math> \displaystyle x_1<x_2 </math>. If the inequality holds strictly <math>\displaystyle (f(x_1)>f(x_2)) </math> | + | Similarlly, a function <math> \displaystyle f:A\to B</math> is called [[monotonically decreasing]] if <math>\displaystyle f(x_1)\geq f(x_2) </math> holds whenever <math> \displaystyle x_1<x_2 </math>. If the inequality holds strictly <math>\displaystyle (f(x_1)>f(x_2)) </math>, |
then the function is called [[strictly decreasing]]. | then the function is called [[strictly decreasing]]. | ||
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====Continuity==== | ====Continuity==== | ||
− | Intuitively, a continuous function has the propriety that | + | Intuitively, a continuous function has the propriety that its graph can be drawn without taking the pencil off the paper. But the reality about continuous function is more complex. |
=====Epsilon-delta definition===== | =====Epsilon-delta definition===== | ||
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=====Heine definition===== | =====Heine definition===== | ||
− | The previous definition of continuity at <math> x_{0} </math> is equivalent with the following: for every sequence <math> (x_n)_{n\geq 0} </math> such that <math> \displaystyle \lim_{n\to\infty}x_n=x_0 </math> we have that <math> \lim_{n\to\infty}f(x_n)=f(x_0) </math>. | + | The previous definition of continuity at <math> x_{0} </math> is equivalent with the following: for every sequence <math> (x_n)_{n\geq 0} </math> such that <math> \displaystyle \lim_{n\to\infty}x_n=x_0 </math>, we have that <math> \lim_{n\to\infty}f(x_n)=f(x_0) </math>. |
It is easy to see that a function is continuous in [[isolated point]]s, and is continuous in [[accumulation point]]s [[iff]] the limit of the function in those point equals the value of the function. | It is easy to see that a function is continuous in [[isolated point]]s, and is continuous in [[accumulation point]]s [[iff]] the limit of the function in those point equals the value of the function. | ||
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=====Intermediate value property===== | =====Intermediate value property===== | ||
− | If a function is continuous then it has the [[Intermediate value property]]. The converse is not always true. | + | If a function is continuous, then it has the [[Intermediate value property]]. The converse is not always true. |
''Proof'':... | ''Proof'':... | ||
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==History of the concept== | ==History of the concept== | ||
− | Without being used explicitly, the notion of function first appears | + | Without being used explicitly, the notion of function first appears with the ancient Greeks and Egyptians. |
− | The | + | The rigorous definition was stated in the 19th century and is the result of the works of some famous mathematicians: [[A.L. Cauchy]], [[L. Euler]], [[B. Riemann]]. With the development of [[set theory]], a new branch of mathematics appeared, [[Mathematical Analysis]], in which the notion of function has a central role. |
==See Also== | ==See Also== |
Revision as of 13:10, 27 June 2006
This article is a stub. Help us out by expanding it.
A function is a rule that maps one set of values to another set of values. For instance, one function may map 1 to 1, 2 to 4, 3 to 9, 4 to 16, and so on. This function has the rule that it takes its input value, and squares it to get an output value. Lets call this function . A common notation to define is: . This tells us that is a function that squares its argument (its input value). Note that this "rule" can be arbitrarily complicated and doesn't need to be given by a simple formula or description. The only requirement is that should be uniquely determined by . The following are examples of functions:
for , otherwise
Contents
[hide]Introductory Topics
Domain and Range
The domain of a function is the set of input values for the argument of a function. The range of a function is the set of output values for that function. For an example, consider the function . The domain of the function is the set , where is a real number. The range is the set of all non-negative real numbers because the square root can never return a negative value.
Real functions
A real function is a function whose range is in the real numbers. Usually we speak about function whose domain is also a subset of the real numbers.
The graph of a function
Functions are often graphed. To find out if a graph is a function, it must stand up to the vertical line test.
The Inverse of a Function
The inverse of a function is a function that "undoes" a function. For an example, consider the function: f(x). The function has the property that . In this case, is called the (right) inverse function. (Similarly, a function so that is called the left inverse function. Typically the right and left inverses coincide on a suitable domain, and in this case we simply call the right and left inverse function the inverse function.) Often the inverse of a function is denoted by .
Intermediate Topics
Injections, surjections, bijections
- An injective function (or one-to-one) is a function which has distinct values for distinct arguments.
By definition, is injective if , or equivalently, .
If and are finite sets, injectivity implies .
Monotonic functions
A function is called monotonically increasing if holds whenever . If the inequality holds strictly , then the function is called strictly increasing.
Similarlly, a function is called monotonically decreasing if holds whenever . If the inequality holds strictly , then the function is called strictly decreasing.
Olympiad and University Level Topics
Functions of Real Variables
A real function is a function whose range is in the real numbers. Usually we speak about function whose domain is also a subset of the real numbers.
Continuity
Intuitively, a continuous function has the propriety that its graph can be drawn without taking the pencil off the paper. But the reality about continuous function is more complex.
Epsilon-delta definition
A function is called continuous at if, for all , there exists such that and .
Heine definition
The previous definition of continuity at is equivalent with the following: for every sequence such that , we have that .
It is easy to see that a function is continuous in isolated points, and is continuous in accumulation points iff the limit of the function in those point equals the value of the function.
A function is continuous on a set if it is continuous in every point of the set.
Properties of continuous functions
- The sum and product of two continuous functions is a continuous function.
- The composition of two continuous functions is a continuous function.
- ...
Intermediate value property
If a function is continuous, then it has the Intermediate value property. The converse is not always true. Proof:...
Continuity on compact intervals
Differentiability
For functions of one variable, differentiablility is simply the question of whether or not a derivative exists. For functions of more than one variable, it's significantly more complicated. In the case of both one and multivariable functions, differentiability implies continuity.
Integrability
Convexity
History of the concept
Without being used explicitly, the notion of function first appears with the ancient Greeks and Egyptians.
The rigorous definition was stated in the 19th century and is the result of the works of some famous mathematicians: A.L. Cauchy, L. Euler, B. Riemann. With the development of set theory, a new branch of mathematics appeared, Mathematical Analysis, in which the notion of function has a central role.