Real Functions and their Graphs

A function whose range is in the real numbers is called real functions.A functional rule can be represented in variety of ways,it is rule that assign to each element in one set (Domain)a unique element in another set (Range). A function whose domain and Co-Domain are subsets of real number are real valued function. Graph of real function can be denoted as G(f), therefore G(f)={(x,f(x)); for all x ←D(f)}
and the set of images or resultant values of f(x) for all x←D(f) is Range of 'f' is denoted by R( x)={f(x); for x←D(f) }Real functions and thier graphs is explained in detail below.

f is a function from set A to a set B if each element x in A can be associated with a unique element in B.

The unique element B which f associates with x in A denoted by f (x).

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Domain is the field from where we chose the value of function.
let f :A→B be a real-valued function, here A is Domain the above definition of the function, 'f' it denoted by D(f)={x:x←R,f(x)←R}. Set A is called domain.D

In the above definition of the function, set B is called co-domain.It contain all the elements of Domain.
Let f:x→Y,f(x)=y
Elements of Domain are Independent variable whereas elements of Range are dependent because Domain finally define what is the Range will.During this relation may be some element of Y are not image of co-domain of X.
Collection of all elements of set Y called Co-Domain.

Domain = { a,b,c,d}
Range = {e,f,g,i}
Co-Domain = {e,f,g,h,i}

A real valued function f : A to B or simply a real function 'f ' is a rule which associates to each possible real number xA, a unique real number f(x)B, when A and B are subsets of R, the set of real numbers.Range contain all the resultant values of the function

In other words, functions whose domain and co-domain are subsets of R, the set of real numbers, are called real valued functions.

If 'f ' is a function and x is an element in the domain of f, then image

f(x) of x under f is called the value of 'f ' at x.
Graph can convey complete information about the function in simple and visual way.
Thus,graph helps for understanding functions and their behaviors

A function f : A ® B Such that A, B Ì R, is said to be a constant function if there exist K Î B such that f(x) = k.

Domain = A

Range = {k}

The graph of this function is a line or line segment parallel to x-axis. Note that, if k>0, the graph B is above X-axis. If k<0, the graph is below the x-axis. If k = 0, the graph is x-axis itself.

A function f : R® R is said to be an identity function if for all x Î R, f(x) = x.

Domain = R

Range = R

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A function f : R® R is said to be a polynomial function if for each x Î R, f(x) is a polynomial in x.

f(x) = x³ + x² + x

f : R ® R such that f(x) = |x|, is called the modulus function or absolute value function.

Domain = R

Since square root of a negative number is not real, we define a function f : R⁺ ® R such that

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For every x←R ,[x] denoted as greatest integer, where [x] ≤ x

if x is an integer then [x] = x

else [x] =n where n is an integer such that
f (x) = [x] = greatest integer less than or equal to x

For a real number x, we denote by [x], the smallest integer greater than or equal to x. For example, [5 . 2] = 6, [-5 . 2] = -5, etc. The function f:RR defined by

f(x) = [x], xR

is called the smallest integer function or the ceiling function.

Domain: R

Range : Z

The exponential function is defined as f(x) = e^x. Its graph is

It has following Properties:
(1) Domain = R, Range = (0,

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If a is any +ve real number,a≠1 then function f(x)=log_ax is called logrithmic function.Logarithmic function is f (x) = log x.D_f=(0,∞)
Its graph is

Trigonometric functions are sinx, cosx, tanx, etc. The graph of these functions have been done in class XI.

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Inverse of function is only possible if function is One -to-one.for the inverse of trigonometric functions restricted there domain to some interval.Since value of sine function lies between [-1,1] at angle [-∏/2,∏/2].By restricting it domain to the interval function is one-to-one at [-∏/2,∏/2] ,so inverse of function is possible.
Inverse functions are sin^-1x, cos^-1x, tan^-1x etc. The graph of these functions have been done in class XI.

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f(x) is called the Signum Function,also denoted by sgn x.Its D_f=R and R_f={-1,0,1}
It is a many-one function.

A function f : AB is said to be an odd function if

f(x) = f(-x) = -f(x) for all xA

The domain and range of f depends on the definition of the function.

Examples of odd function are
y = sinx, y = x³, y = tanx

let f(x) = sinx
f(-x) = sin(-x)
= -sinx
⇒f(-x) = -f(x)

A function f : AB is said to be an even function if

f(x) = f(-x) for all xA.

The domain and range of f depends on the definition of the function.

Examples of even function are

y = cosx, y = x², y = secx

A polynomial with only even powers of x is an even function.

The function f defined by

Example: Sketch the graph of 1/f(x) = 1/(x+5)1/2
Step1: Sketch the graph for function f(x)=(x+5)1/2
Step2: Domain of f(x) is [-5,∞)
Step3: f(x)>0 for all x>-5,f(-5)=0
F is increasing function
Step 4: Use these information to sketch the graph of 1/f(x)
Step 5 : 1/f has Domain (-5,∞)
Step 6: 1/f > 0 on (-5,∞)
1/f is decreasing function and has vertival asymptots at x=-5