kwizNET Subscribers, please login to turn off the Ads!
Email us to get an instant 20% discount on highly effective K-12 Math & English kwizNET Programs!

Online Quiz (Worksheet A B C D)

Questions Per Quiz = 2 4 6 8 10

High School Mathematics - 2
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.
  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.
  • Have your essay responses graded by your teacher and enter your score here!

    Your score for this chapter (0-100%) =

    Subscription to kwizNET Learning System offers the following benefits:

    • Unrestricted access to grade appropriate lessons, quizzes, & printable worksheets
    • Instant scoring of online quizzes
    • Progress tracking and award certificates to keep your student motivated
    • Unlimited practice with auto-generated 'WIZ MATH' quizzes
    • Child-friendly website with no advertisements
    • Choice of Math, English, Science, & Social Studies Curriculums
    • Excellent value for K-12 and ACT, SAT, & TOEFL Test Preparation
    • Get discount offers by sending an email to

    Quiz Timer