Notes On Functions - CBSE Class 11 Maths
Relation A relation R from a non-empty set A to a non-empty set B is a subset of the Cartesian product A × B. Example: Let A = {1,2,3,4} and B = {2,3,4,5,6} and a relation R: A → B defined by R = {(x,y):y=x+1}. R = {(1, 2), (2, 3), (3, 4), (4, 5)} Domain of relation R = A Codomain of relation R = B Range of relation R = {2, 3, 4, 5} Range ⊂ Co-domain Statements Let X be a set of books in a shop. Y is the set of prices that correspond to each of the books in the shop. We have a correspondence between the books and their prices. Let N be the set of natural numbers and C = {1, 8, 27, 64 …………..}. We can assign each natural number to its cube, giving a correspondence between the two sets, N and C. In both the examples, we have two sets and a rule that assigns to every member of the first set a unique element of the second set. Function is a type of a relation that assigns to every member of one set a unique element of another set. Every function is a relation, but every relation is not a function. Example 1: Let A = {1,2,3,4} and B = {2,3,4,5,6}, and a relation R: A → B defined by R = {(x,y):y=x+1}. R = {(1, 2), (2, 3), (3, 4), (4, 5)} Observe that all the elements of set A have a correspondence with the elements of set B. Every element of set A has only one correspondence with the elements of set B. Such type of relations in which every element in the domain set has one and only one image in set B are called functions. Hence, R is a function. Let A and B be two non-empty sets. A function f : AàB, is a rule that maps each member of set A with a unique member of set B. A relation R from a non-empty set A to another non-empty set B is said to be a function if its domain is set A and no two ordered pairs of R have the same first coordinates. Example 2: f = {(4,6), (3,9), (-11,6), (3,11)} (3,9) Î f and (3,11) Î f Observe that both the ordered pairs have the same first coordinates. Hence, the given relation is not a function. Example 3: f = {(x, x): x is a real number} In this relation, for all real numbers, there exists an image that is the same real number. Hence, the given relation is a function. f is relation from set A to set B and f is said to be a function from A to B if, for all x ÎA, there exists a unique y ÎB such that (x, y)Î f . Let A = {1, 2, 3, 4} and B = {2, 3, 4, 5, 6}, and a function f: A → B defined by f = {(x, y): y = x+1} or, f = {(1, 2), (2, 3), (3, 4), (4, 5)} Domain = A and Codomain = B If f: AàB is a function and (x, y) Î f, then y is called the image of x under the function f and we denote this as f(x) = y. Also, x is known as the pre-image or inverse image of y. (1, 2), (2, 3), (3, 4), (4, 5) ∈ f f(1) = 2, 2 is called the image of 1, or 1 is called the pre-image or inverse image of 2. f(2) = 3, 3 is called the image of 2, or 2 is called the pre-image or inverse image of 3. f(3) = 4, 4 is called the image of 3, or 3 is called the pre-image or inverse image of 4. and f(4) = 5, 5 is called the image of 4, or 4 is called the pre-image or inverse image of 5. Range If f: AàB is a function, then the set of all the images of elements of set A is known as the range of the function f. It is denoted by f (A). The range of a function is a subset to the codomain of the function. f(A) = { 2, 3, 4, 5} and f(A) ⊆ B In y = f(x), x is called the independent variable, while y is known as the dependent variable. The dependent variable is also called the value of function f at x. Example: Let f: AàR be a function defined by f(x) = 2x + 1, where A = {1, 2, -1}. f(1) = 2(1) + 1 = 3 f(2) = 2(2) + 1 = 5 f(-1) = 2(-1) + 1 = -1 f = { (1, 3), (2, 5), (-1,-1)} Vertical Line Test The vertical line test helps us to check whether a given relation is a function or not. If the vertical line intersects the graph of a elation at more than one place, it means that two ordered pairs have same first coordinates. Then we can conclude that it is not a function. Algebra of functions A function that has either R (the set of real numbers) or one of its subsets as its range is called a Real Valued Function. If the domain and the range of the function is either R or a subset of R, then it is called a Real Function. Addition of two functions: If f: X à R and g : X à R are two real functions, where X ⊂ R, then f + g: X à R is defined as (f + g)(a) = f(a) + g(a), "a Î X. Example: If f = {(1, 2), (2, 3), (3, 4)} and g = {(1, 5), (2, 6), (3, 7)}, then f(1) = 2 and g(1) = 5, then (f + g)(1)= f(1) + g(1) = 2 + 5 ⇒ (1,2 + 5) ∈ f + g f +g = {( 1, 2+5), (2, 3+6), (3, 4+7)} Þ f +g = {(1, 7), (2, 9), (3, 11)} Subtraction of a real function from another: Let f : X àR and g : X à R be any two real functions, where X ⊂ R. Then we define (f - g) : XàR by (f-g) (a) = f(a) -g(a), for all a Î X. Example: If f = {(1,2), (2,3), (3,4)} and g = {(1,5), (2,6), (3,7)} then (f -g)(1)= f(1)-g(1) = 2-5 ⇒ (1,2 - 5) ∈ f - g f -g = {( 1,2-5). (2,3-6), (3,4-7)} Þ f -g = {( 1, -3). (2, -3), (3, -3)} Multiplication of a real function by a scalar (real number): Let f : XàR be a real valued function and k be a real number. Then the product kf : XàR is defined by (kf) (a) = kf (a), a Î X. Example: If f = {(1,2), (2,3), (3,4)} and k = 2 f(1) = 2, then 2f (1)= 2 x 2 ⇒ (1,2 x 5) ∈ 2f 2f = {(1, 2×2), (2, 2×3), (3, 2×4)} Þ 2f = {(1,4), (2,6), (3,8)} Multiplication of two real functions: The product (or multiplication) of two real functions f : XàR and g : XàR is a function fg : XàR defined by (fg) (a) = f(a) g(a), for all a ÎX. If f = {(1,2), (2,3), (3,4)} and g = {(1,5), (2,6), (3,7)} f(1) = 2 and g(1) = 5, then (f  x g)(1)= f(1) x g(1) = 2 x 5 ⇒ (1,2 x 5) ∈ f x g f g = {( 1,2×5). (2,3×6), (3,4×7)} ⇒ f g = {( 1, 10), (2, 18), (3, 28)} Quotient of two real functions Let f : X àR and g : X à R be any two real functions, where X ⊂ R. Then we define (f/g) : XàR by (f/g) (a) = f(a) /g(a), provided g(a) ¹0, for all a Î X. If f = {(1,2), (2,3), (3,4)} and g = {(1,5), (2,6), (3,7)} f(1) = 2 and g(1) = 5, then (f /g)(1)= f(1)/g(1) = 2/5 ⇒ (1,2 / 5) ∈ f / g f /g = {( 1,2/5), (2,3/6), (3,4/7)} ⇒ f /g = {( 1, 2/5), (2, ½), (3, 4/7)}

#### Summary

Relation A relation R from a non-empty set A to a non-empty set B is a subset of the Cartesian product A × B. Example: Let A = {1,2,3,4} and B = {2,3,4,5,6} and a relation R: A → B defined by R = {(x,y):y=x+1}. R = {(1, 2), (2, 3), (3, 4), (4, 5)} Domain of relation R = A Codomain of relation R = B Range of relation R = {2, 3, 4, 5} Range ⊂ Co-domain Statements Let X be a set of books in a shop. Y is the set of prices that correspond to each of the books in the shop. We have a correspondence between the books and their prices. Let N be the set of natural numbers and C = {1, 8, 27, 64 …………..}. We can assign each natural number to its cube, giving a correspondence between the two sets, N and C. In both the examples, we have two sets and a rule that assigns to every member of the first set a unique element of the second set. Function is a type of a relation that assigns to every member of one set a unique element of another set. Every function is a relation, but every relation is not a function. Example 1: Let A = {1,2,3,4} and B = {2,3,4,5,6}, and a relation R: A → B defined by R = {(x,y):y=x+1}. R = {(1, 2), (2, 3), (3, 4), (4, 5)} Observe that all the elements of set A have a correspondence with the elements of set B. Every element of set A has only one correspondence with the elements of set B. Such type of relations in which every element in the domain set has one and only one image in set B are called functions. Hence, R is a function. Let A and B be two non-empty sets. A function f : AàB, is a rule that maps each member of set A with a unique member of set B. A relation R from a non-empty set A to another non-empty set B is said to be a function if its domain is set A and no two ordered pairs of R have the same first coordinates. Example 2: f = {(4,6), (3,9), (-11,6), (3,11)} (3,9) Î f and (3,11) Î f Observe that both the ordered pairs have the same first coordinates. Hence, the given relation is not a function. Example 3: f = {(x, x): x is a real number} In this relation, for all real numbers, there exists an image that is the same real number. Hence, the given relation is a function. f is relation from set A to set B and f is said to be a function from A to B if, for all x ÎA, there exists a unique y ÎB such that (x, y)Î f . Let A = {1, 2, 3, 4} and B = {2, 3, 4, 5, 6}, and a function f: A → B defined by f = {(x, y): y = x+1} or, f = {(1, 2), (2, 3), (3, 4), (4, 5)} Domain = A and Codomain = B If f: AàB is a function and (x, y) Î f, then y is called the image of x under the function f and we denote this as f(x) = y. Also, x is known as the pre-image or inverse image of y. (1, 2), (2, 3), (3, 4), (4, 5) ∈ f f(1) = 2, 2 is called the image of 1, or 1 is called the pre-image or inverse image of 2. f(2) = 3, 3 is called the image of 2, or 2 is called the pre-image or inverse image of 3. f(3) = 4, 4 is called the image of 3, or 3 is called the pre-image or inverse image of 4. and f(4) = 5, 5 is called the image of 4, or 4 is called the pre-image or inverse image of 5. Range If f: AàB is a function, then the set of all the images of elements of set A is known as the range of the function f. It is denoted by f (A). The range of a function is a subset to the codomain of the function. f(A) = { 2, 3, 4, 5} and f(A) ⊆ B In y = f(x), x is called the independent variable, while y is known as the dependent variable. The dependent variable is also called the value of function f at x. Example: Let f: AàR be a function defined by f(x) = 2x + 1, where A = {1, 2, -1}. f(1) = 2(1) + 1 = 3 f(2) = 2(2) + 1 = 5 f(-1) = 2(-1) + 1 = -1 f = { (1, 3), (2, 5), (-1,-1)} Vertical Line Test The vertical line test helps us to check whether a given relation is a function or not. If the vertical line intersects the graph of a elation at more than one place, it means that two ordered pairs have same first coordinates. Then we can conclude that it is not a function. Algebra of functions A function that has either R (the set of real numbers) or one of its subsets as its range is called a Real Valued Function. If the domain and the range of the function is either R or a subset of R, then it is called a Real Function. Addition of two functions: If f: X à R and g : X à R are two real functions, where X ⊂ R, then f + g: X à R is defined as (f + g)(a) = f(a) + g(a), "a Î X. Example: If f = {(1, 2), (2, 3), (3, 4)} and g = {(1, 5), (2, 6), (3, 7)}, then f(1) = 2 and g(1) = 5, then (f + g)(1)= f(1) + g(1) = 2 + 5 ⇒ (1,2 + 5) ∈ f + g f +g = {( 1, 2+5), (2, 3+6), (3, 4+7)} Þ f +g = {(1, 7), (2, 9), (3, 11)} Subtraction of a real function from another: Let f : X àR and g : X à R be any two real functions, where X ⊂ R. Then we define (f - g) : XàR by (f-g) (a) = f(a) -g(a), for all a Î X. Example: If f = {(1,2), (2,3), (3,4)} and g = {(1,5), (2,6), (3,7)} then (f -g)(1)= f(1)-g(1) = 2-5 ⇒ (1,2 - 5) ∈ f - g f -g = {( 1,2-5). (2,3-6), (3,4-7)} Þ f -g = {( 1, -3). (2, -3), (3, -3)} Multiplication of a real function by a scalar (real number): Let f : XàR be a real valued function and k be a real number. Then the product kf : XàR is defined by (kf) (a) = kf (a), a Î X. Example: If f = {(1,2), (2,3), (3,4)} and k = 2 f(1) = 2, then 2f (1)= 2 x 2 ⇒ (1,2 x 5) ∈ 2f 2f = {(1, 2×2), (2, 2×3), (3, 2×4)} Þ 2f = {(1,4), (2,6), (3,8)} Multiplication of two real functions: The product (or multiplication) of two real functions f : XàR and g : XàR is a function fg : XàR defined by (fg) (a) = f(a) g(a), for all a ÎX. If f = {(1,2), (2,3), (3,4)} and g = {(1,5), (2,6), (3,7)} f(1) = 2 and g(1) = 5, then (f  x g)(1)= f(1) x g(1) = 2 x 5 ⇒ (1,2 x 5) ∈ f x g f g = {( 1,2×5). (2,3×6), (3,4×7)} ⇒ f g = {( 1, 10), (2, 18), (3, 28)} Quotient of two real functions Let f : X àR and g : X à R be any two real functions, where X ⊂ R. Then we define (f/g) : XàR by (f/g) (a) = f(a) /g(a), provided g(a) ¹0, for all a Î X. If f = {(1,2), (2,3), (3,4)} and g = {(1,5), (2,6), (3,7)} f(1) = 2 and g(1) = 5, then (f /g)(1)= f(1)/g(1) = 2/5 ⇒ (1,2 / 5) ∈ f / g f /g = {( 1,2/5), (2,3/6), (3,4/7)} ⇒ f /g = {( 1, 2/5), (2, ½), (3, 4/7)}

Next