Ib Maths Tutors-Types of functions(part-3)

Types of Functions-

IB Maths Tutors should give twenty-two hours for teaching functions and equations as per IBO recommendations. This is my third article on functions in the series of ib mathematics

IB Maths Tutors should give twenty hours in teaching functions and equations.This is my third article on functions in the series of ib mathematics

For the sake of a comprehensive discussion, some standard functions and their graphs are discussed here

For the sake of a comprehensive discussion, some standard functions and their graphs are discussed here.

1.Greatest Integer Function–  This is an interesting function. It is defined as the largest     integer less than or equal to x

                                                         y = [x].

For all real numbers, x, this function gives the largest integer less
than or equal to x.

For example:   [1] = 1      [2.5] = 2      [4.7] = 4      [5.3] = 5
Beware!    [-2] = -2      [-2.6] = -3      [-4.1] = -5      [-6.5] = -7

domain=R
range=Z

<img src="ibmathstutors.jpg" alt="ibmathstutors">

greatest integer function

2. Fractional part function-

for every real value of x this function gives the fractional part of x.

                                                            f(x)={x}

                                            {2.3}=.3 , {5.4}=.4, {2.2}=.2

                                          {6.7}=.7, {-2.3}=.7, {-2.6}=.3

We can say that:                              0≤{x}∠1

domain=R
range=less than 1

<img src="ibmathstutors.jpg" alt="ibmathstutors">

Fractional Part

3. Polynomial function–    These are functions of the form

       f(x) = anxn + an−1x n−1 + . . . + a2x 2 + a1x + a0 .

Constant, linear, quadratic, cubic, quartic functions etc fall in this category
domain of these functions is R and range is either R or a subset of R

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polynomial

 

4. Trigonometric functions-  Trigonometric functions or circular functions draw the relationship between the sides and angles of right triangles .we can find this relationship using “unit circle”. I have explained all this thing in the given video.

There are six trigonometric functions, we will discuss them all one by one
i. Sin function(variation in a)

 f(x)=sin x
this is a periodic function with a period of

domain=R
range=[-1,1]

<img src="ibmathstutors.jpg3" alt="ibmathstutors3">

Sin function

ii. Cosine function(variation in b)-

f(x)=cosx

this is also a periodic function with a period of

domain=R

range=[-1,1]

<img src="ibmathstutors.jpg4" alt="ibmathstutors4">

cosine function

iii. Tangent function(variation in a/b)-                     

                                                                        f(x)= tan x

this is also a periodic function with a period of pie

domain=R-{n pie+pie/2}

range=R

<img src="ibmathstutors.jpg5" alt="ibmathstutors">

tan function

This was my last post in ib maths tutors-function series. In my next post, I will discuss some questions based on these topics.

Classification  Of  Functions :

(i) One – One Function (Injective mapping)-: A function f: A \to B is said to be a one-one function  or injective mapping if different elements of  A have different f  images in B.  Thus for  {x_1},{\rm{ }}{x_2} \in A &,f{(x)_1},{\rm{f(}}{x_2}) \in B {x_1} = {x_2} \Leftrightarrow f({x_1}) = f({x_2}) Function is one-one while if

{x_1} \ne {x_2} \Leftrightarrow f({x_1}) \ne f({x_2}) The function will not be one-one.

(ii) If f(x) is any function which is entirely increasing or decreasing in whole domain, then f(x) is one-one.

(iii) If any line parallel to x-axis cuts the graph of the function atmost at one point, then the function is one-one.

Many–one function-: A function f: A \to B  is said  to be  many one functions  if two or more elements of A have the  same f image in  B. Thus  f: A \to B is  many-one  if

\;{x_1},{x_2} \in A\;\& ,f{(x)_1}{\rm{ = f}}({x_2}) but  {x_1} \ne {x_2}

(i) Any continuous function which has at least one local maximum or local minimum, then f(x) is many-one. In other words,  if a line parallel to x-axis cuts the graph of the function at least at two points, then f is many-one. This test is known as horizontal line test

(ii) If a function is one-one, it cannot be many-one and vice versa.

Onto function (Surjective mapping)-: If the function f: A \to B is such that each element in B (co-domain) is the image of at least one element in A, then we say that f is a function of A ‘onto’ B . Thus f: A \to B is surjective if \forall   b  \in  B,  \exists  some  a  \in  A  such that  f (a) = b

Into function-: If f: A \to B is such that there exists at least one element in co-domain which is not the image of any element in the domain, then f(x) is into.

(i) If a function is onto, it cannot be into and vice versa.

(ii) A polynomial of degree even will always be into.

Thus a function can be one of these four types :

(a) one-one onto (injective & surjective)

(b) one-one into (injective but not surjective)

(c) many-one onto (surjective but not injective)

(d) many-one into (neither surjective nor injective)

Bijective mapping- If f is both injective & surjective, then it is called a Bijective mapping.The bijective functions are also named as invertible,  non-singular or bi-uniform functions. If a  set  A contains n

If a  set  A contains n distinct elements then the number of different functions defined from A \to B is nn & out of it n ! are one one.

Algebraic  Operations  On  Functions: If f & g are real-valued functions of x with domain set A, B respectively, then both f & g are defined in A \cap B  Now we define  f + g,

f – g ,  (f . g) &  (f/g) as follows -:

(i) (f ± g) (x) = f(x) ± g(x)

(ii) (f . g) (x) = f(x) . g(x)

(iii)  \frac{f}{g}(x) = \frac{{f(x)}}{{g(x)}}

Composite Of Uniformly & Non-Uniformly Defined Functions: Let  f :  A \to B  and g : B \to C  be two functions . Then the function gof :  A \to C  defined by (gof) (x) = g (f(x))  \forall

x \in  A is called the composite of the two functions f & g.

Properties  Of  Composite  Functions : (i) The composite of functions is not commutative  i.e.

(i) The composite of functions is not commutative  i.e.  gof  \ne fog .

(ii) The composite of functions is associative  i.e.  if  f, g, h are three functions such that  fo(goh) &  (fog)oh  are defined, then  fo(goh) = (fog)oh

(iii) The composite  of  two bijections is a bijection  i.e.  if  f & g are two bijections such that  gof is defined, then gof is also a bijection.Implicit  &  Explicit

Implicit  &  Explicit Function-: A function defined by an equation not solved for the dependent variable is called an implicit Function. For eg. the equation x3 + y3= 1 defines y  as an implicit function. If y has been expressed in terms of x alone then it is called an Explicit Function.

Homogeneous  Functions-: A function is said to be homogeneous with respect to any set of variables when each of its terms is to the same degree with respect to those variables.  For  example F(x)=  5 x2 + 3 y2 – xy  is  homogeneous  in  x & y . Symbolically if, f (tx , ty) = tn.  f(x,y)  then  f(x,y) is homogeneous function of degree  n.

Inverse  Of  A  Function-: Let  f: A \to  B  be a  one-one  &  onto function,  then  there  exists  a  unique  function   g: B \to A  such that  f(x) = y  \Leftrightarrow g(y) = x, \forall  \;x \in A and \;y \in B

Then g is said to be inverse of f.  Thus  g =f-1  B  \to  A =  {(f(x), x) ½ (x,  f(x)) Î f} . Properties  Of  Inverse  Function  : (i) The inverse of a bijection is unique.

(ii) If f: A \to B  is a bijection & g: A \to A is the inverse of f, then fog =IB  and gof =IA

where  I&  IB  are identity functions on the sets A & B respectively. Note that the graphs of f & g  are the mirror images of each other in the line y = x.

Odd & Even Functions-: If f (-x) = f (x) for all x in the domain of ‘f’ then f is said to be an even function. e.g. f (x) = cos x  ;  g (x) = x² + 3 .

If f (-x) = -f (x) for all x in the domain of ‘f’ then f is said to be an odd function.

e.g. f (x) = sin x , g (x) = x3 + x

(i) f (x) – f (-x) = 0 =>  f (x) is even  &  f (x) + f (-x) = 0 => f (x) is odd

(ii) f (x) – f (-x) = 0 =>  f (x) is even  &  f (x) + f (-x) = 0 => f (x) is odd .

(iii) A function may neither be odd nor be even.

(iv) Inverse  of  an  even  function  is  not  defined .

(v) Every even function is symmetric about the y-axis  &  every odd  function is symmetric about the origin .

(vi) Every function can be expressed as the sum of an even & an odd function.

                            f(x) = \frac{{f(x) + f( - x)}}{2} + \frac{{f(x) - f( - x)}}{2}

(vii) The only function which is defined on the entire number line & is even and odd at the same time is f(x) = 0.(viii) If f and g both  are even or both are odd then the function  f.g  will  be even but if any one of them is odd then f.g  will  be odd .

(viii) If f and g both are even or both are odd then the function  f.g  will be even but if any one of them is odd then f.g  will be odd.

Periodic  Function-: A function  f(x) is  called  periodic  if  there exists a positive number T (T > 0) called the period  of the  function  such  that  f (x + T) = f(x),  for  all  values  of  x within the domain of x.

e.g. The function sin x & cos x both are periodic over 2\pi  & tan x is periodic over \pi

(i) f (T) = f (0) = f (-T) ,   where ‘T’ is the period .

(ii) Inverse of a periodic function does not exist .

(iii) Every constant function is always periodic, with no fundamental period .

(iv) If  f (x)  has  a period  T  &  g (x)  also  has  a  period T  then it does not  mean that        f(x) + g(x)  must have a  period T .   e.g. f(x) = \left| {\sin x} \right| + \left| {\cos x} \right|

(v) If  f(x) has a period  \pi , then    \sqrt {f(x)}   and  \frac{1}{{f(x)}}   also has a period  \pi

(vi) if  f(x) has a period T then f(ax + b) has a period  T/a  (a > 0).

Here are links to my previous posts on functions

First Post-An Introduction to functions

Second Post-Domain and Range of functions

Third Post-Types of functions(part-1)

Here is a pdf containing questions on this topic

 functions .pdf

 

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