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Functions, properties. elementary functions and their inverses. 2. előadás. Function. Video: http://www.youtube.com/user/MyWhyU?v=Imn_Qi3dlns. Function.
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Functions, properties. elementary functions and their inverses 2. előadás
Function Video: http://www.youtube.com/user/MyWhyU?v=Imn_Qi3dlns
Function • A function, denoted by f, is a mapping from a set A to a set Bwhich 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 Dfand 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.
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.
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 .
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 .
Bijection, bijective function Definition: A function is called a bijection, orbijectivefunctionif 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 Eis 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.
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 .
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 DgDf 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
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
Composed function In function composition, you're plugging an entire function for the x: Definition:Given f: XY, g: Y Z; then go 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+5and f o g (x)=f(g(x))= 3(2x)+5
Inverse of(to) a function • Definition: Let fbe 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:
Linear function transformation Transforming the variable Transforming the functional value
Transforming the variable The graph is translatedby –c alongthe x axis
Transforming the variable If 0<a<1 If a<1
Transforming the variable The left side of axis y is neglected, and the right hand side of y is reflected o axis y
Transforming the functional value The graph is translated along the y axis, if c is positive, then to + direction, if -, then to the - direction
Transforming the functional value Graph is reflected to the x axis
Transforming the functional value 1<a 1<a 0<a<1
Transforming the functional value The negative part of the graph is reflected to the x axis
Function classification Powerfunctions
Functionclassification Polinomials
Function classification Rationalfunctions
Function classification Irrationalfunctions: ifitsequationconsistsalso a fractioninapower
Function classification Exponentialfunction: ax
Logarithmic functions based of.. Function classification where
Function classification Trigonometri(cal) functions
Elementary functions:Power, exponentional, trigonometrical and their inverses, and functions of their +,*,/
Bounded 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: A function can have an upper bound, lower bound, both or be unbounded. 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 unbounded if it is not bounded above or below. A function is bounded if it is bounded above and below.
Increasing and Decreasing Functions 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 Ask: what is y doing? as you read from left to right. Increasing Decreasing Write your answer in set theory in terms of x
Monotonity and inverse If the funcion is strictly monoton, then it has an inverse
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.
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.
Point of inflexion A point on the graph of a function where the curve changes concavity is called an inflection point.
Concave down= Concave + - Going from positive to negative means its decreasing. • 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.
When the tangent slopes are increasing the graph of f is concave up. Concavity Going from negative to positive means its increasing. + - Concave up=convex
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)
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.
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.).
