functions, properties. elementary functions and their inverses

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Functions, properties. elementary functions and

their inverses

2. előadás

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Function

Video:

http://www.youtube.com/user/MyWhyU?v=Imn_Qi3dlns

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Function

A function, denoted by f, is a mapping from a set A to a set B which sarisfies the following:for each element a in A, there is an element b in B. The set A in the above definition is called the Domain of the function Df and B its codomain. The Range (or image) of the function Rf is a subset of a codomain. Thus, f is a function if it covers the domain (maps every element of the domain) and it is single valued.

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Vertical lines test

If we have a graph of a function in a usual Descartes coordinate system, then we can decide easily whether a mapping is a function or not:

it is a function if there are no vertical lines that intersect the graph at more than one point.

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Injective function

A function f is said to be one-to-one (injective) , if and only if whenever f(x) = f(y) , x = y .

Example: The function f(x) = x2 from the set of natural numbers N to N is a one-to-one function. Note that f(x) = x2 is not one-to-one if it is from the set of integers(negative as well as non-negative) to N , because for example f(1) = f(-1) = 1 .

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Surjective function

A function f from a set A to a set B is said to be onto(surjective) , if and only if for every element y of B , there is an element x in A such that f(x) = y , that is, f is onto if and only if f( A ) = B .

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Bijection, bijective functionDefinition: A function is called a bijection , or bijective function if it is onto and one-to-one.

Example: The function f(x) = 2x from the set of natural numbers N to the set of non-negative even numbers E is an onto function. However, f(x) = 2x from the set of natural numbers N to N is not onto, because, for example, nothing in N can be mapped to 3 by this function.

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Bijection, bijective function

Horizontal Line Test: A function f is one to one iff its graph intersects every horizontal line at most once.

If f is either an increasing or a decreasing function on its domain, then is one-to-one .

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Restriction, extension

Sometime we have to restrict or extend the original domain of a function.

That is, that we keep the mapping, but the domain of the function is a subset of the original domain: function g is a restriction of function f, if Dg Df and g(x)=f(x). Function f is the extension of g.

Example: f(x)= x2 Df =R. g(x)= x2 Dg=R+

f is not bijective, function g is bijective

x

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Operations on fuctions

Let f and g be functions from a set A to the set of real numbers R.

Then the sum , the product , and the quotient of f and g are defined as follows: - for all x, ( f + g )(x) = f(x) + g(x) , and - for all x, ( f*g )(x) = f(x)*g(x) , f(x)*g(x) is the product of two real numbers f(x) and g(x). - for all x, except for x-es where g(x)=0, ( f/g )(x) = f(x)/g(x) ( f/g )(x) is a quotient of two real numbers f(x) and g(x)

Example: Let f(x) = 3x + 1 and g(x) = x2 . Then ( f + g )(x) = x2 + 3x + 1 , and ( f*g )(x) = 3x3 + x2 =h(x), if l(x)=x, then (h/l)(x)=3 x2 +x

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Composed function

In function composition, you're plugging an entire function for the x:

Definition:Given f: XY, g: Y Z; then g o f: X Z is defined by

g o f(x) = g(f(x)) for all x.

Read “g composed with f” or “g circle of f”, or “g’s of f” )

Example: f(x)=3x+5, g(x) = 2x then

g o f (x)= g(f(x)= 23x+5 and f o g (x)=f(g(x))= 3(2x)+5

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Inverse of(to) a function

Definition: Let f be a function with domain D and range R. A function g with domain R and range D is an inverse function for f if, for all x in D,

y = f(x) if and only if x = g(y). Examples:

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Linear function transformation

Transforming the variable Transforming the functional value

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Transforming the variable

The graph is translated by –c along the x axis

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If 0<a<1If a<1

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The left side of axis y is neglected, and the right hand side of y is reflected o axis y

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Transforming the functional value

The graph is translated along the y axis, if c is positive, then to + direction, if -, then to the - direction

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Graph is reflected to the x axis

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1<a 1<a

0<a<1

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The negative part of the graph is reflected to the x axis

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Function classification

Power functions

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Polinomials

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Rational functions

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Irrational functions: if its equation consists also a fraction in a power

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Exponential function: ax

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Function classificationLogarithmic functions based of..

where

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Trigonometri(cal) functions

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Elementary functions:Power, exponentional, trigonometrical and their inverses, and functions of their +,*,/

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Bounded: A function can have an upper bound, lower bound, both or be unbounded.

Bounded above: if there is a number B such that B is greater than or equal to every number in the range of f. (think maximum)

Bounded below: if there is a number B such that B is less than or equal to every number in the range of f. (think minimum)

A function is bounded if it is bounded above and below.

A function is unbounded if it is not bounded above or below.

Bounded

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Let x1 and x2 be numbers in the domain of a function, f.

The function f is increasing over an open interval if for every x1 < x2 in the interval, f(x1) < f(x2).

The function f is decreasing over an open interval if for every x1 < x2 in the interval, f(x1) > f(x2).

Increasing and Decreasing Functions

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Ask: what is y doing? as you read from left to right.

Write your answer in set theory in terms of x

Increasing

( , 5) (0,3) (6, ) Decreasing

( 5,0) (3,6)

Increasing and Decreasing Functions

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Monotonity and inverse

If the funcion is strictly monoton, then it has an inverse

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Global minima, maxima

Suppose that a is in the domain of the function f such that, for all x in the domain of f,

f(x) < f(a) then a is called a maximum of f. Suppose that a is in the domain of the

function f such that, for all x in the domain of f,

f(x) > f(a) then a is called a minimum of f.

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Local minima and maxima

Suppose that a is in the domain of the function f and suppose that there is an open interval I containing a which is contained in the domain of f such that, for all x in I,

f(x) < f(a) then a is called a local maximum of f. Suppose that a is in the domain of the function f and

suppose that there is an open interval I containing a which is also contained in the domain of f such that, for all x in I,

f(x) > f(a) then a is called a local minimum of f.

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Where are local and global maximas,minimas?

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A point on the graph

of a function where

the curve changes

concavity is called

an inflection point.

Point of inflexion

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• If f ”(x) < 0 on an interval (a, b) then f ’ is decreasing on that interval.

When the tangent slopes are decreasing the graph of f is concave down.

Concave down=

Concave

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Concavity

When the tangent slopes are increasing the graph of f is concave up.

Concave up=convex

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PARITY OF FUNCTIONS

A function is "even" when:

f(x) = f(-x) for all x (symmetrical around y)

A function is "odd" when:

-f(x) = f(-x) for all x (symmetrical around the origin)

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Graphs of some even functions

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Graphs of some odd functions

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Special Properties of odd and even functions

Adding: The sum of two even functions is even The sum of two odd functions is odd The sum of an even and odd function is neither even nor odd

(unless one function is zero). Multiplying: The product of two even functions is an even function. The product of two odd functions is an even function. The product of an even function and an odd function is an odd

function.

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Periodic functions In mathematics, a periodic function is a function that

repeats its values in regular intervals or periods. A function is said to be periodic (or, when

emphasizing the presence of a single period instead of multiple periods, singly periodic) with period if

for , 2, .... For example, the sine function , illustrated above, is periodic with least period (often simply called "the" period) (as well as with period , , , etc.).

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Inverse of sine: arc sin x

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Inverse of cosine: arc cos x

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Inverse of tan: arc tg x

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Inverse of cotan: arc ctg x

functions, properties. elementary functions and their inverses

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