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, assigning to each value in the first set exactly one value in the second. 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. One can call this function <math>f</math>. | |
− | + | ==Rigorous Definition== | |
+ | Let <math>A</math>,<math>B</math> be [[set]]s and let <math>f</math> be a [[subset]] of <math>A\times B</math>, which denotes the [[Cartesian product]] of <math>A</math> and <math>B</math>. (That is, <math>f</math> is a [[relation]] between <math>A</math> and <math>B</math>.) We say that <math>f</math> is a ''function from <math>A</math> to <math>B</math>'' (written <math>f: A \to B</math>) if and only if | ||
− | <math> f( | + | * For every <math>a \in A</math> there is some <math>b \in B</math> such that <math>(a, b) \in f</math>, and |
+ | * if <math>(a,b)\in f</math> and <math>(a,c)\in f</math> then <math>b=c</math>. (Here <math>(a,b)</math> is an [[ordered pair]].) | ||
− | <math>f(x)=\ | + | ==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: <math>f(x) = \sqrt{x^2-9}</math>. The domain of the function is the set <math>\{x:|x| \geq 3\}</math>, where <math>x</math> is a real number, because the square root is only defined when its argument is nonnegative. 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 functions whose domain is also a [[subset]] of the real numbers. | ||
− | + | ===Graphs=== | |
+ | Functions are often graphed. A graph corresponds to a function only if it stands up to the [[vertical line test]]. | ||
− | <math>g(x)= | + | ===Inverses=== |
+ | The inverse of a function is a function that "undoes" a function. For an example, consider the function: <math>f(x) = x^2 + 6</math>. The function <math>g(x) = \sqrt{x-6}</math> has the property that <math>f(g(x)) = x</math>. Therefore, <math>g</math> is called the '''(right) inverse function'''. (Similarly, a function <math>g</math> that satisfies <math>g(f(x))=x</math> 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 <math>f</math> is denoted by <math>f^{-1}</math>. | ||
− | + | ==Intermediate Topics== | |
− | == | + | ===Injections, Surjections, Bijections=== |
+ | *An [[injection]] (or one-to-one function) is a function which always gives distinct values for distinct arguments within a given domain. | ||
+ | **By definition, <math>f:A\to B</math> is injective if <math>f(a)=f(b) \Rightarrow a=b </math>, or equivalently, <math>a\neq b \Rightarrow f(a)\neq f(b).</math> | ||
+ | **Injectivity of a function <math>f: A \rightarrow B</math> implies that <math>f</math> has an inverse. Furthermore, if <math>A</math> and <math>B</math> are finite sets, injectivity implies <math>|A|\leq |B|</math>. | ||
+ | *A [[surjection]] (or onto function) maps at least one element from its domain, <math>A,</math> onto every element of its range, <math>B.</math> | ||
+ | *A [[bijection]] (or one-to-one correspondence, which must be one-to-one and onto) is a function, <math>f: A \rightarrow B</math> that is both injective and surjective. | ||
+ | **If <math>f</math> is an injection from <math>A \rightarrow B</math> and <math>g</math> is an injection from <math>B \rightarrow A,</math> then there exists a bijection, <math>h,</math> between <math>A</math> and <math>B</math>. This is the [[Schroeder-Bernstein Theorem]]. | ||
− | == | + | ====Examples==== |
− | == | + | * <math>f(x) = x</math> is injective and surjective (and therefore bijective) from <math>\mathbb{C} \rightarrow \mathbb{C}</math>. |
− | + | * <math>f(x) = x^2</math> is injective from <math>\mathbb{R^+} \rightarrow \mathbb{R}</math>. | |
− | == | + | * <math>f(x) = x^2</math> is surjective from <math>\mathbb{R} \rightarrow \mathbb{R^+}</math>. |
− | + | * <math>f(x) = x^2</math> is neither injective from <math>\mathbb{R} \rightarrow \mathbb{R}</math> (since <math>f(2) = 4 = f(-2)</math>) nor surjective from <math>\mathbb{R} \rightarrow \mathbb{R}</math> (since <math>f</math> does not map any value to <math>-5</math>, which is an element of <math>\mathbb{R}</math>). | |
===Monotonic functions=== | ===Monotonic functions=== | ||
− | A function <math> f:A\to B</math> is called [[monotonically increasing]] if <math>f(x_1)\geq f(x_2) </math> | + | A function <math>f:A\to B</math> is called [[monotonically increasing]] if <math>f(x_1)\geq f(x_2) </math> holds whenever <math>x_1>x_2 </math>. If the inequality holds strictly <math>(f(x_1)>f(x_2)) </math>, |
then the function is called [[strictly increasing]]. | then the function is called [[strictly increasing]]. | ||
− | + | Similarly, a function <math>f:A\to B</math> is called [[monotonically decreasing]] if <math>f(x_1)\geq f(x_2) </math> holds whenever <math>x_1<x_2 </math>. If the inequality holds strictly <math>(f(x_1)>f(x_2)) </math>, | |
− | then the function is called [[strictly decreasing]]. | + | then the function is called [[strictly decreasing]]. |
+ | |||
+ | ==Advanced 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=== | ===Continuity=== | ||
+ | Intuitively, a continuous function has the propriety that its graph can be drawn without taking the pencil off the paper. To rigorously define continuous functions, more complex mathematics is necessary. | ||
+ | |||
+ | ====Epsilon-Delta Definition==== | ||
+ | A function <math>f:E\to\mathbb{R}</math> is called ''continuous'' at some point in its domain <math> x_{0} </math> if, for all <math> \varepsilon >0 </math>, there exists <math>\delta >0</math> such that, for any <math>x \in E</math>, the condition <math> |x-x_0|<\delta </math> implies that <math>|f(x)-f(x_0)|<\varepsilon </math>. | ||
+ | |||
+ | ====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>\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 large groups of points [[if]] the limit of the function in those points 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 are continuous functions. | ||
+ | * The composition of two continuous functions is a continuous function. | ||
+ | * In any closed interval <math>[a, b]</math>, there exist real numbers <math>c</math> and <math>d</math> such that <math>f</math> has a [[maximum]] value at <math>c</math> and <math>f</math> has a [[minimum]] value at <math>d</math>. | ||
+ | * If a function is continuous, then it has the [[Intermediate Value Theorem]]. The converse is not always true. | ||
+ | |||
===Differentiability=== | ===Differentiability=== | ||
+ | Differentiability is a smoothness condition on functions. For functions of one variable, differentiability is simply the question of whether or not a [[derivative]] exists. For functions of more than one variable, the notion of differentiability is significantly more complicated. In the case of both one and multivariable functions, differentiability implies continuity. | ||
+ | |||
+ | A single-variable function <math>f(x)</math> is differentiable at <math>x=a</math> if <math>\lim_{x\rightarrow a} \frac{f(x)-f(a)}{x-a} \in \mathbb{R}</math>. The derivative is the value of this [[limit]]. | ||
+ | |||
===Integrability=== | ===Integrability=== | ||
+ | All continuous functions are integrable, as well as many non-continuous functions. | ||
+ | |||
===Convexity=== | ===Convexity=== | ||
+ | A twice-differentiable function <math>f(x) </math> is concave up (or ''convex'') in the interval <math>[a,b] </math> if <math>f''(x)>0</math> in the interval <math>[a,b] </math> and concave down (or ''concave'') if <math>f''(x)<0 </math>. The points of inflection, when the concavity switches, of the function occur at the roots of <math>f''(x) </math>. | ||
+ | |||
+ | ==Notation== | ||
+ | A common notation to define <math>f</math> is: <math>f(x) = x^2</math> (where the <math>x^2</math>, of course, is merely an example). 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>f(x)=x ^ {2}+2x-2</math> | ||
+ | |||
+ | *<math>f(x)=\sin(\log{x})</math> | ||
+ | |||
+ | *<math>f(x)=x^2</math> for <math>x>0</math>, otherwise <math>f(x)= \sin{x}</math> | ||
+ | |||
+ | *<math>f(x)=p(g(x))</math> | ||
+ | |||
+ | *<math>g(x)=F'(x)</math> | ||
+ | |||
+ | == History of Functions == | ||
+ | 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]], [[Leonhard Euler]], and [[Bernhard 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. | ||
+ | |||
+ | The current notation used is attributed to [[Leonhard Euler]]. | ||
+ | |||
+ | ==Problems== | ||
+ | ===Introductory=== | ||
+ | *Define <math>x\otimes y=x^3-y</math>. What is <math>h\otimes (h\otimes h)</math>? | ||
+ | |||
+ | <math> \mathrm{(A) \ } -h\qquad \mathrm{(B) \ } 0\qquad \mathrm{(C) \ } h\qquad \mathrm{(D) \ } 2h\qquad \mathrm{(E) \ } h^3 </math> | ||
+ | ([[2006 AMC 10A Problems/Problem 2|Source]]) | ||
+ | ===Intermediate=== | ||
+ | *The [[function]] f is defined on the [[set]] of [[integer]]s and satisfies | ||
+ | <math> | ||
+ | f(n)= | ||
+ | \begin{cases} | ||
+ | n-3 & \mbox{if }n\ge 1000 \\ | ||
+ | f(f(n+5)) & \mbox{if }n<1000 | ||
+ | \end{cases} | ||
+ | </math> | ||
+ | Find <math>f(84)</math>. | ||
+ | ([[1984 AIME Problems/Problem 7|Source]]) | ||
+ | |||
+ | ===Olympiad=== | ||
+ | *Let <math>f</math> be a function with the following properties: | ||
+ | # <math>f(n)</math> is defined for every positive integer <math>n</math>; | ||
+ | # <math>f(n)</math> is an integer; | ||
+ | # <math>f(2)=2</math>; | ||
+ | # <math>f(mn)=f(m)f(n)</math> for all <math>m</math> and <math>n</math>; | ||
+ | # <math>f(m)>f(n)</math> whenever <math>m>n</math>. | ||
+ | |||
+ | Prove that <math>f(n)=n</math>. | ||
+ | ([[1969 Canadian MO Problems/Problem 8|Source]]) | ||
+ | |||
+ | ===Advanced=== | ||
+ | *Describe all polynomials <math>P(x)</math> such that <math>P(x + 1) - 1 = P(x) + P'(x)</math> for all <math>x</math>. | ||
+ | |||
+ | (<url>weblog_entry.php?t=182628 Source</url>) | ||
+ | |||
+ | ==See Also== | ||
+ | *[[Odd function]] | ||
+ | *[[Even function]] | ||
+ | * [[Algebra]] | ||
+ | ** [[Functional equation]] | ||
+ | ** [[Polynomials]] | ||
+ | * [[Calculus]] | ||
+ | ** [[Limit]] | ||
+ | ** [[Derivative]] | ||
+ | ** [[Integral]] | ||
+ | |||
+ | [[Category:Algebra]] | ||
+ | [[Category:Definition]] | ||
+ | [[Category:Functions]] |
Latest revision as of 03:16, 12 May 2023
A function is a rule that maps one set of values to another set of values, assigning to each value in the first set exactly one value in the second. 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. One can call this function .
Contents
Rigorous Definition
Let ,
be sets and let
be a subset of
, which denotes the Cartesian product of
and
. (That is,
is a relation between
and
.) We say that
is a function from
to
(written
) if and only if
- For every
there is some
such that
, and
- if
and
then
. (Here
is an ordered pair.)
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, because the square root is only defined when its argument is nonnegative. 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 functions whose domain is also a subset of the real numbers.
Graphs
Functions are often graphed. A graph corresponds to a function only if it stands up to the vertical line test.
Inverses
The inverse of a function is a function that "undoes" a function. For an example, consider the function: . The function
has the property that
. Therefore,
is called the (right) inverse function. (Similarly, a function
that satisfies
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 injection (or one-to-one function) is a function which always gives distinct values for distinct arguments within a given domain.
- By definition,
is injective if
, or equivalently,
- Injectivity of a function
implies that
has an inverse. Furthermore, if
and
are finite sets, injectivity implies
.
- By definition,
- A surjection (or onto function) maps at least one element from its domain,
onto every element of its range,
- A bijection (or one-to-one correspondence, which must be one-to-one and onto) is a function,
that is both injective and surjective.
- If
is an injection from
and
is an injection from
then there exists a bijection,
between
and
. This is the Schroeder-Bernstein Theorem.
- If
Examples
is injective and surjective (and therefore bijective) from
.
is injective from
.
is surjective from
.
is neither injective from
(since
) nor surjective from
(since
does not map any value to
, which is an element of
).
Monotonic functions
A function is called monotonically increasing if
holds whenever
. If the inequality holds strictly
,
then the function is called strictly increasing.
Similarly, a function is called monotonically decreasing if
holds whenever
. If the inequality holds strictly
,
then the function is called strictly decreasing.
Advanced 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. To rigorously define continuous functions, more complex mathematics is necessary.
Epsilon-Delta Definition
A function is called continuous at some point in its domain
if, for all
, there exists
such that, for any
, the condition
implies that
.
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 large groups of points if the limit of the function in those points 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 are continuous functions.
- The composition of two continuous functions is a continuous function.
- In any closed interval
, there exist real numbers
and
such that
has a maximum value at
and
has a minimum value at
.
- If a function is continuous, then it has the Intermediate Value Theorem. The converse is not always true.
Differentiability
Differentiability is a smoothness condition on functions. For functions of one variable, differentiability is simply the question of whether or not a derivative exists. For functions of more than one variable, the notion of differentiability is significantly more complicated. In the case of both one and multivariable functions, differentiability implies continuity.
A single-variable function is differentiable at
if
. The derivative is the value of this limit.
Integrability
All continuous functions are integrable, as well as many non-continuous functions.
Convexity
A twice-differentiable function is concave up (or convex) in the interval
if
in the interval
and concave down (or concave) if
. The points of inflection, when the concavity switches, of the function occur at the roots of
.
Notation
A common notation to define is:
(where the
, of course, is merely an example). 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
History of Functions
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, Leonhard Euler, and Bernhard 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.
The current notation used is attributed to Leonhard Euler.
Problems
Introductory
- Define
. What is
?
(Source)
Intermediate
Find
.
(Source)
Olympiad
- Let
be a function with the following properties:
is defined for every positive integer
;
is an integer;
;
for all
and
;
whenever
.
Prove that .
(Source)
Advanced
- Describe all polynomials
such that
for all
.
(<url>weblog_entry.php?t=182628 Source</url>)