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A relation *f * from a set A to set B is said to be **function** if every element of set A has one and only one image in set B.

A function *f * is a relation if no two ordered pairs in the relation have the same first element.

f = {(1, 2), (2, 3), (3, 4), (4, 5), (5, 6), (6, 7)}

A function that is a relation between two sets can be represented graphically because it is the set of ordered pairs and can be plotted on the Cartesian plane.

The notation *f* : A → B means that *f* is a function from A to B.

A =Domain of *f* = {1, 2, 3, 4, 5, 6} * *and B = Co-domain of *f* = {2, 3, 4, 5, 6, 7}

Given *a* ϵ A, there is a unique element *b* in B, such that *f* (*a*) = b.

The set of all values of *f* (*a*) taken together is called the **range of f** or the image of A under

Symbolically, range of

A function f:R → R is said to be an identity function if f(x) = x, ∀ x ϵ R denoted by I

Let A = {1, 2, 3}

The function f: A → A defined by f(x) = x is an identity function.

f = {(1,1), (2,2), (3,3)}.

The graph of an identity function is a straight line passing through the origin.

Each point on this line is equidistant from the coordinate axes.

The straight line makes an angle of 45° with the coordinate axes.

**Linear function**

A function f:R → R is said to be a linear function if f (x) = ax + b, where a ≠ 0, a and b are real constants, and x is a real variable.

Consider the linear function, f(x) = 3x + 7, ∀ x ϵ R

The ordered pairs satisfying the linear function are (0, 7), (-1, 4), (-2, 1).

On plotting these points on the Cartesian plane and then joining them, we get the graph of the linear function f of x is equal to 3x + 7, ∀ x ϵ R as shown in the figure.

**Constant function**

A function *f*:R → R is said to be a constant function, if f(x) = c , ∀ x ϵ R, where c is a constant.

Let *f*:R → R be a constant function defined by f(x) = 4 , ∀ x ϵ R.

The ordered pairs satisfying the linear function are: (0, 4), (-1, 4), (2, 4).

If the range of a function is a singleton set, then it is known as a constant function.

On plotting these points on the Cartesian plane and then joining them, we get the graph of the constant function f of x = 4,∀ x ϵ R as shown.

Also, from the graph, we can conclude that the graph of a constant function, f(x) = c, is always a straight line parallel to the X-axis, intersecting the Y-axis at (0, c).

**Polynomial function**

A function f:R → R is said to be a polynomial function if for all x in R, y = f(x) = a_{0} + a_{1}x + a_{2}x^{2} + ……………..+ a_{n}x^{n}, where n is a non-negative integer and a_{0}, a_{1}, a_{2……..} a_{n} are real numbers.

**Examples of polynomial functions**

f(x) = x^{2} + 5x + 6 ,∀ x ϵ R

and f (x) = x^{3} + 4x + 2 ,∀ x ϵ R

Note: In a polynomial function, the powers of the variables should be non-negative integers.

For example, f(x) = √x + 2 (∀ x ϵ R) is not a polynomial function because the power of x is a rational number.

Consider the polynomial function, f(x) = 3x^{2} +2x -3 ,∀ x ϵ R

The ordered pairs satisfying the polynomial function are (0, -3), (-1, -2), (1, 2), (2, 13), (-2, 5).

On plotting these points on the Cartesian plane and then joining them, we get the graph of the polynomial function f of x is equal to 3 x^{2} + 2 x - 3 ,∀ x ϵ R as shown.

**Rational function**

If f(x) and g(x) be two polynomial functions, then $\frac{\text{f(x)}}{\text{g(x)}}$ such that g(x) ≠ 0 and ∀ x ϵ R, is known as a rational function.

Let us consider the function f ( x) = $\frac{\text{2x - 5}}{\text{3x - 2}}$ (x ≠ $\u2154$).

The ordered pairs satisfying the polynomial function are: (0, 5), (2, -¼), (1, -3).

On plotting these points on the Cartesian plane and then joining them, we get the graph of the given rational function as shown.

**Modulus function**

A function f: R → R defined by f(x) = |x| (∀ x ϵ R) is known as a modulus function.

If x is negative, then the value of the function is minus x, and if x is non-negative, then the value of the function is x. i.e. f(x) = x if x ≥ 0 = - x if x < 0.

The ordered pairs satisfying the polynomial function are (0, 0), (-1, 1), (1, 1), (-3, 3), (3, 3).

On plotting these points on the Cartesian plane and then joining them, we get the graph of modulus function f of x is equal to mod of x.

**Signum function**

A function f: R → R defined by f(x) = $\frac{\left|\text{x}\right|}{\text{x}}$ is called a signum function.

It can also be defined as

1 if x > 0

f(x) = 0 if x = 0

-1 if x < 0 (,∀ x ϵ R)

The ordered pairs satisfying the signum function are (1, 1), (2, 1), (0, 0), (-1, -1), (-5, -1).

On plotting these points on the Cartesian plane and then joining them, we get the graph of signum function f of x is equal to mod of x divided by x.

The domain of the signum function is **R,** and the range is {-1,0,1}.

It is also written as sign (x).

**Greatest integer function**

A function f: R → R defined by f(x) = [x],∀ x ϵ R assumes the value of the greatest integer, less than or equal to x.

From the definition of [x], we can see that

[x] = -1 for -1 £ x < 0

[x] = 0 for 0 £ x < 1

[x] = 1 for 1 £ x < 2

[x] = 2 for 2 £ x < 3, and so on.

⇒ f(2.5) will give the value 2 and f(1.2) will give the value 1, and so on…

Hence, the graph of the greatest integer function is as shown.

A relation *f * from a set A to set B is said to be **function** if every element of set A has one and only one image in set B.

A function *f * is a relation if no two ordered pairs in the relation have the same first element.

f = {(1, 2), (2, 3), (3, 4), (4, 5), (5, 6), (6, 7)}

A function that is a relation between two sets can be represented graphically because it is the set of ordered pairs and can be plotted on the Cartesian plane.

The notation *f* : A → B means that *f* is a function from A to B.

A =Domain of *f* = {1, 2, 3, 4, 5, 6} * *and B = Co-domain of *f* = {2, 3, 4, 5, 6, 7}

Given *a* ϵ A, there is a unique element *b* in B, such that *f* (*a*) = b.

The set of all values of *f* (*a*) taken together is called the **range of f** or the image of A under

Symbolically, range of

A function f:R → R is said to be an identity function if f(x) = x, ∀ x ϵ R denoted by I

Let A = {1, 2, 3}

The function f: A → A defined by f(x) = x is an identity function.

f = {(1,1), (2,2), (3,3)}.

The graph of an identity function is a straight line passing through the origin.

Each point on this line is equidistant from the coordinate axes.

The straight line makes an angle of 45° with the coordinate axes.

**Linear function**

A function f:R → R is said to be a linear function if f (x) = ax + b, where a ≠ 0, a and b are real constants, and x is a real variable.

Consider the linear function, f(x) = 3x + 7, ∀ x ϵ R

The ordered pairs satisfying the linear function are (0, 7), (-1, 4), (-2, 1).

On plotting these points on the Cartesian plane and then joining them, we get the graph of the linear function f of x is equal to 3x + 7, ∀ x ϵ R as shown in the figure.

**Constant function**

A function *f*:R → R is said to be a constant function, if f(x) = c , ∀ x ϵ R, where c is a constant.

Let *f*:R → R be a constant function defined by f(x) = 4 , ∀ x ϵ R.

The ordered pairs satisfying the linear function are: (0, 4), (-1, 4), (2, 4).

If the range of a function is a singleton set, then it is known as a constant function.

On plotting these points on the Cartesian plane and then joining them, we get the graph of the constant function f of x = 4,∀ x ϵ R as shown.

Also, from the graph, we can conclude that the graph of a constant function, f(x) = c, is always a straight line parallel to the X-axis, intersecting the Y-axis at (0, c).

**Polynomial function**

A function f:R → R is said to be a polynomial function if for all x in R, y = f(x) = a_{0} + a_{1}x + a_{2}x^{2} + ……………..+ a_{n}x^{n}, where n is a non-negative integer and a_{0}, a_{1}, a_{2……..} a_{n} are real numbers.

**Examples of polynomial functions**

f(x) = x^{2} + 5x + 6 ,∀ x ϵ R

and f (x) = x^{3} + 4x + 2 ,∀ x ϵ R

Note: In a polynomial function, the powers of the variables should be non-negative integers.

For example, f(x) = √x + 2 (∀ x ϵ R) is not a polynomial function because the power of x is a rational number.

Consider the polynomial function, f(x) = 3x^{2} +2x -3 ,∀ x ϵ R

The ordered pairs satisfying the polynomial function are (0, -3), (-1, -2), (1, 2), (2, 13), (-2, 5).

On plotting these points on the Cartesian plane and then joining them, we get the graph of the polynomial function f of x is equal to 3 x^{2} + 2 x - 3 ,∀ x ϵ R as shown.

**Rational function**

If f(x) and g(x) be two polynomial functions, then $\frac{\text{f(x)}}{\text{g(x)}}$ such that g(x) ≠ 0 and ∀ x ϵ R, is known as a rational function.

Let us consider the function f ( x) = $\frac{\text{2x - 5}}{\text{3x - 2}}$ (x ≠ $\u2154$).

The ordered pairs satisfying the polynomial function are: (0, 5), (2, -¼), (1, -3).

On plotting these points on the Cartesian plane and then joining them, we get the graph of the given rational function as shown.

**Modulus function**

A function f: R → R defined by f(x) = |x| (∀ x ϵ R) is known as a modulus function.

If x is negative, then the value of the function is minus x, and if x is non-negative, then the value of the function is x. i.e. f(x) = x if x ≥ 0 = - x if x < 0.

The ordered pairs satisfying the polynomial function are (0, 0), (-1, 1), (1, 1), (-3, 3), (3, 3).

On plotting these points on the Cartesian plane and then joining them, we get the graph of modulus function f of x is equal to mod of x.

**Signum function**

A function f: R → R defined by f(x) = $\frac{\left|\text{x}\right|}{\text{x}}$ is called a signum function.

It can also be defined as

1 if x > 0

f(x) = 0 if x = 0

-1 if x < 0 (,∀ x ϵ R)

The ordered pairs satisfying the signum function are (1, 1), (2, 1), (0, 0), (-1, -1), (-5, -1).

On plotting these points on the Cartesian plane and then joining them, we get the graph of signum function f of x is equal to mod of x divided by x.

The domain of the signum function is **R,** and the range is {-1,0,1}.

It is also written as sign (x).

**Greatest integer function**

A function f: R → R defined by f(x) = [x],∀ x ϵ R assumes the value of the greatest integer, less than or equal to x.

From the definition of [x], we can see that

[x] = -1 for -1 £ x < 0

[x] = 0 for 0 £ x < 1

[x] = 1 for 1 £ x < 2

[x] = 2 for 2 £ x < 3, and so on.

⇒ f(2.5) will give the value 2 and f(1.2) will give the value 1, and so on…

Hence, the graph of the greatest integer function is as shown.