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 $f$.

Rigorous Definition

Let $A$,$B$ be sets and let $f$ be a subset of $A\times B$, which denotes the Cartesian product of $A$ and $B$. (That is, $f$ is a relation between $A$ and $B$.) We say that $f$ is a function from $A$ to $B$ (written $f: A \to B$) if and only if

  • For every $a \in A$ there is some $b \in B$ such that $(a, b) \in f$, and
  • if $(a,b)\in f$ and $(a,c)\in f$ then $b=c$. (Here $(a,b)$ 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: $f(x) = \sqrt{x^2-9}$. The domain of the function is the set $\{x:|x| \geq 3\}$, where $x$ 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: $f(x) = x^2 + 6$. The function $g(x) = \sqrt{x-6}$ has the property that $f(g(x)) = x$. Therefore, $g$ is called the (right) inverse function. (Similarly, a function $g$ that satisfies $g(f(x))=x$ 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 $f$ is denoted by $f^{-1}$.

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, $f:A\to B$ is injective if $f(a)=f(b) \Rightarrow a=b$, or equivalently, $a\neq b \Rightarrow f(a)\neq f(b).$
    • Injectivity of a function $f: A \rightarrow B$ implies that $f$ has an inverse. Furthermore, if $A$ and $B$ are finite sets, injectivity implies $|A|\leq |B|$.
  • A surjection (or onto function) maps at least one element from its domain, $A,$ onto every element of its range, $B.$
  • A bijection (or one-to-one correspondence, which must be one-to-one and onto) is a function, $f: A \rightarrow B$ that is both injective and surjective.

Examples

  • $f(x) = x$ is injective and surjective (and therefore bijective) from $\mathbb{C} \rightarrow \mathbb{C}$.
  • $f(x) = x^2$ is injective from $\mathbb{R^+} \rightarrow \mathbb{R}$.
  • $f(x) = x^2$ is surjective from $\mathbb{R} \rightarrow \mathbb{R^+}$.
  • $f(x) = x^2$ is neither injective from $\mathbb{R} \rightarrow \mathbb{R}$ (since $f(2) = 4 = f(-2)$) nor surjective from $\mathbb{R} \rightarrow \mathbb{R}$ (since $f$ does not map any value to $-5$, which is an element of $\mathbb{R}$).

Monotonic functions

A function $f:A\to B$ is called monotonically increasing if $f(x_1)\geq f(x_2)$ holds whenever $x_1>x_2$. If the inequality holds strictly $(f(x_1)>f(x_2))$, then the function is called strictly increasing.

Similarly, a function $f:A\to B$ is called monotonically decreasing if $f(x_1)\geq f(x_2)$ holds whenever $x_1<x_2$. If the inequality holds strictly $(f(x_1)>f(x_2))$, 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 $f:E\to\mathbb{R}$ is called continuous at some point in its domain $x_{0}$ if, for all $\varepsilon >0$, there exists $\delta >0$ such that, for any $x \in E$, the condition $|x-x_0|<\delta$ implies that $|f(x)-f(x_0)|<\varepsilon$.

Heine Definition

The previous definition of continuity at $x_{0}$ is equivalent with the following: for every sequence $(x_n)_{n\geq 0}$ such that $\lim_{n\to\infty}x_n=x_0$, we have that $\lim_{n\to\infty}f(x_n)=f(x_0)$.

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 $[a, b]$, there exist real numbers $c$ and $d$ such that $f$ has a maximum value at $c$ and $f$ has a minimum value at $d$.
  • 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 $f(x)$ is differentiable at $x=a$ if $\lim_{x\rightarrow a} \frac{f(x)-f(a)}{x-a} \in \mathbb{R}$. 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 $f(x)$ is concave up (or convex) in the interval $[a,b]$ if $f''(x)>0$ in the interval $[a,b]$ and concave down (or concave) if $f''(x)<0$. The points of inflection, when the concavity switches, of the function occur at the roots of $f''(x)$.

Notation

A common notation to define $f$ is: $f(x) = x^2$ (where the $x^2$, of course, is merely an example). This tells us that $f$ 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 $f(x)$ should be uniquely determined by $x$. The following are examples of functions:

  • $f(x)=x ^ {2}+2x-2$
  • $f(x)=\sin(\log{x})$
  • $f(x)=x^2$ for $x>0$, otherwise $f(x)= \sin{x}$
  • $f(x)=p(g(x))$
  • $g(x)=F'(x)$

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 $x\otimes y=x^3-y$. What is $h\otimes (h\otimes h)$?

$\mathrm{(A) \ } -h\qquad \mathrm{(B) \ } 0\qquad \mathrm{(C) \ } h\qquad \mathrm{(D) \ } 2h\qquad \mathrm{(E) \ } h^3$ (Source)

Intermediate

$f(n)= \begin{cases}  n-3 & \mbox{if }n\ge 1000 \\  f(f(n+5)) & \mbox{if }n<1000 \end{cases}$ Find $f(84)$. (Source)

Olympiad

  • Let $f$ be a function with the following properties:
  1. $f(n)$ is defined for every positive integer $n$;
  2. $f(n)$ is an integer;
  3. $f(2)=2$;
  4. $f(mn)=f(m)f(n)$ for all $m$ and $n$;
  5. $f(m)>f(n)$ whenever $m>n$.

Prove that $f(n)=n$. (Source)

Advanced

  • Describe all polynomials $P(x)$ such that $P(x + 1) - 1 = P(x) + P'(x)$ for all $x$.

(<url>weblog_entry.php?t=182628 Source</url>)

See Also

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