1. Index FAQ Functions, properties. elementary functions and their inverses 2. előadás

2. Index FAQ Function Video: http://www.youtube.com/user/MyWhyU?v=Imn_Qi3dlns

3. Index FAQ 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 D f and B its codomain. The Range (or image) of the function R f 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.

4. Index FAQ 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.

5. Index FAQ 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) = x 2 from the set of natural numbers N to N is a one-to-one function. Note that f(x) = x 2 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.

6. Index FAQ 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.

7. Index FAQ Bijection, bijective function Definition: 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.

8. Index FAQ 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.

9. Index FAQ 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)= x 2 Df =R. g(x)= x 2 Dg=R + f is not bijective, function g is bijective

10. Index FAQ 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) = x 2. Then ( f + g )(x) = x 2 + 3x + 1, and ( f*g )(x) = 3x 3 + x 2 =h(x), if l(x)=x, then (h/l)(x)=3 x 2 +x

11. Index FAQ Composed function In function composition, you're plugging an entire function for the x: Definition:Given f: X  Y, 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) = 2 x then g o f (x)= g(f(x)= 2 3x+5 and f o g (x)=f(g(x))= 3(2 x )+5

12. Index FAQ 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:

13. Index FAQ Linear function transformation Transforming the variableTransforming the functional value

14. Index FAQ Transforming the variable The graph is translated by –c along the x axis

15. Index FAQ Transforming the variable

16. Index FAQ Transforming the variable If 0<a<1 If a<1

17. Index FAQ Transforming the variable The left side of axis y is neglected, and the right hand side of y is reflected o axis y

18. Index FAQ Transforming the functional value The graph is translated along the y axis, if c is positive, then to + direction, if -, then to the - direction

19. Index FAQ Transforming the functional value Graph is reflected to the x axis

20. Index FAQ Transforming the functional value 1<a 0<a<1

21. Index FAQ Transforming the functional value The negative part of the graph is reflected to the x axis

22. Index FAQ Function classification Power functions

23. Index FAQ Function classification Polinomials

24. Index FAQ Function classification Rational functions

25. Index FAQ Function classification Irrational functions: if its equation consists also a fraction in a power

26. Index FAQ Function classification Exponential function: a x

27. Index FAQ Function classification Logarithmic functions based of.. where

28. Index FAQ Function classification Trigonometri(cal) functions

29. Index FAQ Elementary functions: Power, exponentional, trigonometrical and their inverses, and functions of their +,*,/

30. Index FAQ 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

31. Index FAQ Let x 1 and x 2 be numbers in the domain of a function, f. The function f is increasing over an open interval if for every x 1 < x 2 in the interval, f(x 1 ) < f(x 2 ). The function f is decreasing over an open interval if for every x 1 f(x2). Increasing and Decreasing Functions

32. Index FAQ Ask: what is y doing? as you read from left to right. Write your answer in set theory in terms of x Increasing Decreasing Increasing and Decreasing Functions

33. Index FAQ Monotonity and inverse If the funcion is strictly monoton, then it has an inverse

34. Index FAQ 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.

35. Index FAQ 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.

36. Index FAQ Where are local and global maximas,minimas?

37. Index FAQ A point on the graph of a function where the curve changes concavity is called an inflection point. Point of inflexion

38. Index FAQ If f ” ( x ) < 0 on an interval ( a, b ) then f ’ i s decreasing on that interval. When the tangent slopes are decreasing the graph of f is concave down. Concave down= Concave

39. Index FAQ Concavity When the tangent slopes are increasing the graph of f is concave up. Concave up=convex

40. Index FAQ 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)

41. Index FAQ Graphs of some even functions

42. Index FAQ Graphs of some odd functions

43. Index FAQ 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.

44. Index FAQ Periodic functions In mathematics, a periodic function is a function that repeats its values in regular intervals or periods.mathematicsfunction A function is said to be periodic (or, when emphasizing the presence of a single period instead of multiple periods, singly periodic) with period iffunctionsingly periodic 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.).sineleast periodperiod

45. Index FAQ Inverse of sine: arc sin x

46. Index FAQ Inverse of cosine: arc cos x

47. Index FAQ Inverse of tan: arc tg x

48. Index FAQ Inverse of cotan: arc ctg x