Functions, properties.
elementary functions and
their inverses
2. előadás
Index
FAQ
Function
Video:
http://www.youtube.com/user/MyWhyU?v=Imn_Qi3dlns
Index
FAQ
Function
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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.
Index
FAQ
Vertical lines test
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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.
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) = 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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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 .
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.
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FAQ
Bijection, bijective function
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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 .
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.
x
Example: f(x)= x2 Df =R. g(x)= x2 Dg=R+
f is not bijective, function g is bijective
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) = 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
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) = 2x then
g o f (x)= g(f(x)= 23x+5 and f o g (x)=f(g(x))= 3(2x)+5
Index
FAQ
Inverse of(to) a function
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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:
Index
FAQ
Linear function transformation
Transforming the variable Transforming the functional
value
Index
FAQ
Transforming the variable
The graph is translated by –c along the x axis
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FAQ
Transforming the variable
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FAQ
Transforming the variable
If 0<a<1
If a<1
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FAQ
Transforming the variable
The left side of axis y is neglected, and the right
hand side of y is reflected o axis y
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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
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FAQ
Transforming the functional value
Graph is reflected to the x axis
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FAQ
Transforming the functional value
1<a
1<a
0<a<1
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FAQ
Transforming the functional value
The negative part of the graph is reflected to the x axis
Index
FAQ
Function classification
Power
functions
Index
FAQ
Function classification
Polinomials
Index
FAQ
Function classification
Rational functions
Index
FAQ
Function classification
Irrational functions: if its
equation consists also a
fraction in a power
Index
FAQ
Function classification
Exponential function: ax
Index
FAQ
Function classification
Logarithmic functions based of..
where
Index
FAQ
Function classification
Trigonometri(cal) functions
Index
FAQ
Elementary functions:
Power, exponentional, trigonometrical and their
inverses, and functions of their +,*,/
Index
FAQ
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.
Index
FAQ
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).
Index
FAQ
Increasing and Decreasing Functions
Ask: what is y doing? as you
read from left to right.
Increasing
(, 5)  (0,3)  (6, )
Decreasing
(5,0)  (3,6)
Write your answer in set
theory in terms of x
Index
FAQ
Monotonity and inverse
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If the funcion is strictly
monoton, then it has an inverse
Index
FAQ
Global minima, maxima
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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.
Index
FAQ
Local minima and maxima
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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.
Index
FAQ
Where are local and global
maximas,minimas?
Index
FAQ
Point of inflexion
A point on the graph
of a function where
the curve changes
concavity is called
an inflection point.
Index
FAQ
Concave down=
Concave
• 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.
Index
FAQ
Concavity
When the tangent slopes are increasing the
graph of f is concave up.
Concave up=convex
Index
FAQ
PARITY OF FUNCTIONS
A function is "even" when:
f(x) = f(-x) for all x (symmetrical around y)
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A function is "odd" when:
-f(x) = f(-x) for all x (symmetrical around
the origin)
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Index
FAQ
Graphs of some even functions
Index
FAQ
Graphs of some odd functions
Index
FAQ
Special Properties of odd and
even functions
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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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Index
FAQ
Periodic functions
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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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Index
FAQ
Inverse of sine: arc sin x
Index
FAQ
Inverse of cosine: arc cos x
Index
FAQ
Inverse of tan: arc tg x
Index
FAQ
Inverse of cotan: arc ctg x
Index
FAQ