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High School Mathematics - 2
4.5 Onto and Into Functions

The mapping f: A -> B is called an onto function if the set B is entirely used up, i.e if every element of B is the image of atleast one element of A.
For every b B there exists atleast one element a Asuch that f(a) = b

If function f: A -> B is not onto, that is some of the elements of B remain unmated, then f is called an into function.
Remarks:
  1. An onto function is also called surjective function or a surjection.
  2. A one-one function may be both onto and into.
  3. If A and B are finite sets and f: A->B is surjective, then n(B) <= n(A).
  4. If n(A) = m, n(B) = m, then the possible number of possible surjective mappings from A to B is m!.
: Let A = {-2,-2, 3,3}, B = {4,9} and f: A ->B be a function defined by f(x) = x2, then f is onto because f(-2) = 4, f(-3) = 9, f(3) = 9 i.e f(A) = {4,9} = B.
  • If A and B are finite sets and f: A ->B is surjective, then n(B) <= n(A).
  • If n(A) = m and n(B) - m, then the possible number of possible surjective mappings frm A to B is m!. Example: Let A = {-2, 2, -3, 3}, B = {4,9} and f: A -> B be a function defined by f(x) = x2, then f is onto, because f(-2) and f(2) = 4 and f(-3) and f(3) = 9,i.e f(A) = {4,9} = B

    A function f: N ->N defined by f(x) = 7x is not an onto function, because f(N) = {7,14,21,..} is not equal to N(co-domain).
    Method to find surjectivity of s function
    f: R -> R defined by f(x) = x3 + 5 for all x R.
    1. Choose any arbitrary element y in B.
    2. Put f(x) = y
    3. Solve the equation f(x) = y for x and obtain x in terms of y. Let x = g(y).
    4. If for all values of y B, the values of x obtained from x = g(y) are in A, then f is onto. If there are some y B for which x, given by x = g(y), is not in A, then f is not onto.
    Solution: Let there be an arbitrary element in R. Then f(x) = y = x3 + 5
    x = (y -5)1/3.
    Now for all y R, (y-5)1/3 is a real number. So for all y R (co-domain), there exists x = (y - 5)1/3 in R (domain) such that f(x) = x3 + 5

    Here f: R -> R is an onto function.


    Directions: Answer the following.
  • Q 1: If A = {-2,2,-3,3}, B = {4,9} and f: A->B be a function defined by f(x) = x2, what function is this?
    One-One
    Not a function
    Onto

    Q 2: find the numbers x and y if (x+3, y-5) = (5,0)
    x = 2, y = 5
    x = 3, y = 2
    x = 2, y = 3

    Q 3: Given A = {-2, -1, 0, 1, 2} and B = {-3, -1, 1, 5}. list the elements of S = {(x, y): y = 2x2 - 3, x A and y B}. Is S a function? If so classify it.
    Answer:

    Q 4: {(1, 2), (2, 4), (5, 2), (3, 10), (4, 10)}, is this a function?
    no
    yes

    Q 5: An onto function is also called _____.
    injective
    bijective
    surjective

    Q 6: {(x, y) : y = 5x - 6 for all x belongs to W}
    yes
    no

    Q 7: {(x, y) : y = 5x - 6 for all x belongs to W}. If this is a function, what type of function is this?
    Onto
    One-one
    Not a function

    Q 8: A function is said to be bijective if it is both _____.
    equal and constant
    if it has an inverse
    one-one and onto

    Question 9: This question is available to subscribers only!

    Question 10: This question is available to subscribers only!


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