FUNCTION

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• March 6, 2023
• Courses, Dial Jio, Dial Me, Education

Function : A function from X to Y is defined as a relation X C Y such that no two different ordered pairs of the relation have the same first component and every element of X has an image in Y.

It is denoted by f : X ® Y

 

or X ®C        Y.

 

DOMAIN, RANGE AND CO-DOMAIN OF FUNCTION

Domain : Domain of a function is the set of values of a, when (a,b) belongs to the function.

Range : Range of a function is the set of value of b, when (a, b) belongs to the function.

Co-domain : If (a, b) belong to a function f : A ® B then b is called co-domain of

the function. Range is a subset of co-domain have the same elements.

TYPES OF FUNCTIONS

1. Constant Function : A function ‘f’ defined as f(x) = c, for all x Î R, where c is a real number, is known as a constant function. The domain of a constant function is R and its range is (c).
2. Identity Function : A function that associates to each real number x, the same number x is known as the ‘identity function’ such as f(x) = x for all x in

The domain of identity function is R and its range is also R.

3. Polynomial Function : The function f defined by f(x) = a + a x + a x2 + …a xn

0            1               2                        n

where a0,a1,a2 … an are all real number and n is a non-negative integer is known as a polynomial function.

The domain of polynomial function is R and its range is also R.

4. Reciprocal function : The function

 

f defined by f(x) = 1

x

range is R – {0}.

is known as reciprocal function. Its domain is R – {0} and

 

 

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5. Exponential Function : The function ‘f’ defined by f(x) = a^x, where ‘a’ is any positive real number and x is any real number is known as general exponential function with base ‘a’.

If base is e, then the function f(x) = e^x for all x Î R is known as natural exponen- tial function. Its domain is R and its range is {0, a}.

 

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6. Logarithmic Function : The inverse of exponential function is known as a logarith- mic function. It takes the form

y = log x, a > a, a = l and are defined for x > 0 for a < l, y = log x, is steadily de- creasing for a > l, y = log x, is steadily increasing

 

A particular case of a > l is the function y = loge x = log x

The domain of the logarithmic function is the set of positive real numbers. Hence

the logarithmic function is not defined for zero and negative real numbers, log, x or simply log x is called natural logarithm.

 

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Properties of logarithmic function

• The domain is the set of positive real numbers and range is R

 

• Log

el = 0 and

loge^a = 1

 

• Logexy = logex + logey, x > 0 and > 0 x
• Loge———- = logex – logey, x > 0and y > 0

y

 

• Loge

(x^y) = y log x, x > 0 loga

 

e

• Logax =—— , x > 0

loge

• a^x = e^x log^a
• x = elogx, x > 0

7. Rational Function : A function that can be expressed as the quotient of two polynomial functions is known as a rational It is of the form F(x)

p(x)

=

Q(x)

Q(x) = 0

8.  Irrational Function

It takes the form y = ^2kÖx , K Î N

• Even root index : The function symmetric graph with respect to x-axis.

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• Odd root function : It takes the form y = ^2k+1Öx, K Î N

It is defined over all the real axis and its graph passes through the origin and is

symmetrical in relation to the origin itself. f(- x) = – f(x)

• Square root function : The function f(x) = Öx is called a square root function, where Öx is the positive square root of x. Its domain is the set of all non-negative real

9. The greatest Integer Function : For every x Î R, x is the greatest integer £ x, [x] = x, if x is an integer and [x] is equal to the integer immediately to the left of x if x is an integer and [x] is equal to the integer immediately to the left of x if x is not an
10. Inverse Function : If f be a one-one function with domain D and range R, then a function g : R ® D defined by g(y) = x where f(x) = y is called inverse of f. It is denoted by f-1.

A function f is known as inversible or invertible if it is one-one.

11. Modulus Function : f(x) = ½x½ is the modulus function, if ½x½ denotes the absolute value of x e.

 

x if x is ³ 0

½x½ =—————— The domain of

-x if x is <          ½x½ is R

12. Signum Function : The function f defined by

½x½

f(x) = ——. n ¹ 0

x

0,x = n 1 if x > 0

y = 0 if x = 0 —   is known as

-1 if x < 0

signum function. Its domain is R and its range is {-1, 0, 1}

13. Inverse Trigometric Functions : The real functions sin-1 x, cos-1 x, etc. are known as inverse trignometric functions. The domain and range of these functions are as given in table below :

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14. Monotone Function : A function that is either increasing or decreasing is known as a monotonic function and a function that is either strictly increasing or strictly decreasing is known as a strictly monotonic
15. The function f : X ® Y is called an into function, if there is at least one

element of set Y which has no pre-image in set X.

 

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16. The function f : X — Y is called an onto function if every element of set Y has at least one pre-image in set

 

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17.  The function f : X—- Y is called one-one if distinct elements have distinct

images.

 

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18. The function : X — Y is called many-to-one, if one or more elements of set X there correspond only one element of set

 

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Note : 1. One-one is also written as 1 – 1.

2. An onto function is also called ‘surjection’.
3. A 1 – 1 onto function is called a ‘bijection’.

Representation of a function : A function is represented by :

• verbal description
• an arrow diagram
• in tabular

x 0 1 2 3

The table

y 1 4 5 7

represents a function.

• By a formula known as equation. The equation y = 3x + 5 represents a
• Set builder notation, such as f : {(x,y)} : y = 3x + 5}

How to test for function : It can be tested whether a given relative is a function or not by using the following tests :

• In case of a function, the first set i.e., the domain is fully used up.
• In a function, the first members of all the ordered pairs are
• In a function, each element of the first set has only one image in the second
• In a function, a vertical line will intersect the graph of the function at one point only as shown below :

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In case, “If it is possible to draw a vertical line that intersects the graph of a relation in more than one points, then the relation is not a function otherwise the rela- tion is a function.”

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