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# Functions

Function is a relation between output and input.Output depends upon input. As the input of a function changes the output also changes.Let a function is f(a) = b then a is the input and b is the output. Set of all input is domain and set of all output is range or co-domain.

Function is defined as follows, "function is a relation between domain and range. Considering two sets A and B. We form the Cartesian Product, we form relations. From all the relations, we can select a few which satisfy the rule that each element of the set A is related to only one element of the set B. When a relation satisfies this rule, it is called a function."

In this chapter, we will study how a function is a relation, but a relation may not be a function. Hence, the function calculator sections helps to differentiate relation and a function. If f is a function from A → B and defined as f(a) = b then a = A and b = f(a) = B is unique. A is domain and B is co domain or we can call it as Range of function f. Also, we can say that f from A to B is a relation and every relation from A to B is not a function.

Note: f is a set of ordered pairs, no two of which have the same first coordinate.

The general notation of a function is y = f(x). It is also denotes as f : X → Y where, f is the function defined between X and Y where, x is in X and y is in Y.

 Related Calculators Calculator Functions Calculate Exponential Function Calculate Inverse Function calculating gamma function

## Characteristics of Functions

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Below are the characteristics of functions:

1. Two functions f,g are equal if and only if they have same domain and range"> domain and range and f(x) = g(x) for every x = A
2. f: A ---> B is one-one(Injection) if and only if for all a0 ,a1 = A, “f(a0)=f(a1) implies that a0 = a1
3. f: A ---> B is onto(Surjection) if and only if f(A) = B. also for every b = B, there exists an element a = A such that f(a) = b.
4. f: A ---> B is Bijection if f is one-one and f is onto.then number of elements in A is equal to number of elements in B.

## Types of Functions

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There are some basic types of functions

### Injective Functions:

It is also called as one to one function. It satisfies the property that if f(a) = f(b) then, a = b. This is the condition for injective function

### Surjective Functions:

Surjective function is also called as onto function. Here, every elements in the range is associated with atleast one element in the domain. If y is element in range then there is a x such that y = f(x).

### Bijective Functions:

If the given function is both injective and surjective or one to one and onto then, it is called as a bijective function.

### Inverse Functions:

If f is a function from A to B then inverse of a function is the function defined from B to A.It is denoted by f -1. That is, if f:A -> B then f -1 : B -> A. If f(x) = y then x = f-1(y).

### Solved Examples

Question 1: Check whether the given function is bijective or not f(x)= x + 2
Solution:

f(x) = x + 2

f(a) = a + 2

f(b) = b + 2

If f(a) = f(b), a + 2 = b + 2

a = b

Therefore, it is one to one

y = f(x)

y is directly dependent on x. So, for every value of y there is a x value. Hence, it is onto. So, given function is Bijective.

Question 2: Find the inverse of the function f(x) = 2x + 1
Solution:

f(x) = 2x + 1

y = 2x + 1

2x = y - 1

x = (y - 1)/2

Inverse function is f-1(x) = (x - 1)/2

Question 3: Is the given function surjective f(x) = 2x + 1 where, f is function from R to R
Solution:

y = f(x)

y = 2x + 1

For all real values of x, we get a real value of y. So, for every y, there is a x associated with it. So, the given function is onto or surjective.

Question 4: f(x) = x2 + 2.Show that it is one-one.
Solution:

f(x) = x2 + 2

f(a) = a2 + 2

f(b) = b2 + 2

f(a) = f(b)

a2 + 2 = b2 + 2

a2 = b2 (then take square root on both side)

a = b

Hence, it is one to one or injective.

## Functions of Several Variables

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A function of one variable has one independent variable and one dependent variable. Both the domain and range of functions of one variable are subsets of set of real numbers R. Vector valued functions have subsets of real numbers as domain and their range is a set of vectors in two or more dimensions. Functions of several variables are defined on multidimensional domains and their ranges consist of subsets of R

Real valued functions of several independent variables are defined in a manner similar to the way functions of one variable are defined. Their domains are ordered pairs/triples/n-tuples depending upon the number of variables present.The output or the dependent variable ranges over subsets of real numbers.

## Local and Global Behavior of a Function

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The global behavior includes the characteristics at both ends of the interval. The end values are considered along with the local values for extrema conditions. For example, the greatest among the end values and the local maxima values are considered as the absolute maximum and is called as the global maximum of the function in the given interval. Similarly, the least among the end values and the local minima values are considered as the absolute minimum and is called as the global minimum of the function in the given interval.

## Composition of Functions

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There is one particular way to combine functions.The value of a function f depends upon the value of another variable x and that variable could be equal to another function g, so its value depends on the value of a third variable. If this is the case, then the first variable which is a function h, is called as the composition of two functions(f and g). It denoted as f o g = (f o g) x = f(g(x))

For example
, let f(x) = x+1 and g(x) = 2x, then h(x) = f(g(x)) = f(2x) = 2x + 1.

## Transformations of Functions

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With a little bit change in the given function or basic formula of given function, we can transform or translate any function. There are many ways to transform the given function like, horizontal, vertical, compression, stretch etc. We can show all these things in the graph of the function.

If we have a function y = f(x) = x2, then the graph of this is as follows: Then, we can transform it as:
(1) If we can add some quantity say a in it then the graph is moving up and moving down as per if a > 0 and a < 0 respectively.
Let f(x) = y = x2 + 2 (2) If we add some constant in the value of x as y = f(x) = (x + C)2 then, the graph is moving right and left as per C < 0 and C > 0 respectively. (3) If we multiply the whole function by some constant, then we can able to stretch or compress the graph of given function in the y direction.
f(x) = 2x2 If C > 1, then stretch shown in the graph and if 0 < C < 1 compress in the graph.

If we multiply x by some constant constant, then we can able to stretch or compress the graph of given function in the x direction.
f(x) = (0.5x)2 If C > 1, then compress shown in the graph and if 0 < C < 1 stretch in the graph.

If we multiply the whole function by -1 then, we can able to do upside down the whole function.
f(x) = -x2 ## Generating Functions

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In mathematics, we can use generating functions for the counting problems. If we have a sequence of real numbers in a way that a0, a1, a2,...., an, then we can define a power series for this sequence as $G(x) = a_{0} + a_{1}x + a_{2}x^{2} + a_{3}x^{3} + ......$

where, G(x) is is called the generating function for the given sequence a0, a1, a2,...., an.

### Solved Example

Question: Find the generating function for given sequence and solve it if possible. {an} = 1 + 1 + 1 + 1 + 1 + 1 + 0 + 0 + 0 + 0 +.....
Solution:
The generating function for this is
G(x) = 1 + 1.x + 1.x2 + 1.x3 + 1.x4 + 1.x5 + 0.x6 + 0.x7+ ................
= 1 + x + x2 + x3 + x4 + x5
The above series is in the form of geometric series whose sum is $\frac{1 - x^{6}}{1 - x}$.

## Graphing Functions

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In mathematics, we can draw a graph of any function, but we need some information about that function. If f(x) = y is a function then we have to know about the values of x and y. If we have the values of x then we can easily find out the values of y corresponding values of x. Then, we can make a tabular data of these value. After that put the x values on x-axis and y values on y-axis on the graph and draw a rough sketch of the given function.

### Solved Example

Question: If f(x) = y = (x - 1)3 then, draw the graph of the function.
Solution:
We have f(x) = y = (x - 1)3. Then, first we have to find the values of x and y.
For this let
x = -1, then y = -8
x = 0, then y = -1
x = 1, then y = 0
x = 2, then y = 1
x = 3, then y = 8
Now, by the use of the data, draw the graph of the given function f(x) = y = (x - 1)3 ## Examples of Functions

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Given below are some of the examples on Functions.

### Solved Examples

Question 1:

The arrow diagram represents a relation from set A to set B.

1. Represent the relation in roster form.
2. Is this relation a function? Give reason for your answer.

Solution: 1. {(1, 2), (2, 4), (5, 2), (3, 10), (4, 10)}
2. Yes it is a function. Every first element is mapped and it is many-one. The function is many-one into.

Question 2:

Let f be defined by f(x) find

(i) f(2) (ii) (iii) (iv) x when .

Solution:
1. [f(2) means find the value of y when x = 2].
2. 3. 4. Therefore, 2x + 1 = 3x

x = 1

Question 3:

Which of the following relations are functions:

1. {(x,y) : y $\geq$ (x - 3), x belongs to Z}
2. {(x,y) : x = 3 for all values of y}
3. {(x, y) : y = 4 for all values of x}
4. {(x, y) : y = 5x - 6, x belongs to W}

Solution:
1. Not a function: For one value of x there are many y values and for one value of 'y' there are many 'x' values.
2. Not a function: For one value of x there are many values of y (one - many).
3. Function: Many values of x equal to one value of y (many - one).
4. Function: For one value of x there is one value of y (one - one).

 More topics in Functions Types of Functions Period of a Function Domain and Range of a Function Limit of a Function Vertical Line Test Graphing Functions Explicit Function Implicit Function Combinations of Functions
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