Notes On Functions and their Graphs - CBSE Class 11 Maths
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 f. Symbolically, range of f = {b: b= f (a), for some a in A} = {2, 3, 4, 5, 6, 7}. Different types of functions and their graphical representation. Identity function A function f:R → R is said to be an identity function if f(x) = x, ∀ x ϵ R denoted by IR. 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) = a0 + a1x + a2x2 + ……………..+ anxn, where n is a non-negative integer and a0, a1, a2…….. an are real numbers. Examples of polynomial functions f(x) = x2 + 5x + 6 ,∀ x ϵ R and f (x) = x3 + 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) = 3x2 +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 x2 + 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 ≠ $⅔$). 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.

#### Summary

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 f. Symbolically, range of f = {b: b= f (a), for some a in A} = {2, 3, 4, 5, 6, 7}. Different types of functions and their graphical representation. Identity function A function f:R → R is said to be an identity function if f(x) = x, ∀ x ϵ R denoted by IR. 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) = a0 + a1x + a2x2 + ……………..+ anxn, where n is a non-negative integer and a0, a1, a2…….. an are real numbers. Examples of polynomial functions f(x) = x2 + 5x + 6 ,∀ x ϵ R and f (x) = x3 + 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) = 3x2 +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 x2 + 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 ≠ $⅔$). 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.

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