Q C
Show that the function f : R∗ ∗∗ ∗∗ → R∗ ∗∗∗∗ defined by f (x) = x is one-one and onto, where R∗∗∗ ∗∗ is the set of all non-zero real numbers. Is the result true, if the domain R∗ ∗∗∗∗ is replaced by N with co-domain being same as R∗ ∗∗∗∗?
Let A = R - {3} and B = R - {1}. Consider the function f : A → B defined by f (x) = 2 x x − − . Is f one-one and onto? Justify your answer .
Let f : R → R be defined as f(x) = x 4. Choose the correct answer . (A) f is one-one onto (B) f is many-one onto (C) f is one-one but not onto (D) f is neither one-one nor onto.
Let f : R → R be defined as f (x) = 3x. Choose the correct answer . (A) f is one-one onto (B) f is many-one onto (C) f is one-one but not onto (D) f is neither one-one nor onto. MATHEMA TICS12
Check the injectivity and surjectivity of the following functions: (i) f : N → N given by f (x) = x2 (ii) f : Z → Z given by f(x) = x2 (iii) f : R → R given by f (x) = x2 (iv) f : N → N given by f (x) = x3 (v) f : Z → Z given by f(x) = x3
Prove that the Greatest Integer Function f : R → R, given by f (x) = [x], is neither one-one nor onto, where [x] denotes the greatest integer less than or equal to x.
Show that the Modulus Function f : R → R, given by f (x) = | x |, is neither one- one nor onto, where | x | is x, if x is positive or 0 and | x | is - x, if x is negative.
Show that the Signum Function f : R → R, given by f x x x x ( ) / , / , / , = > = < 1 0 0 0 1 0 if if if is neither one-one nor onto.
Let A = {1, 2, 3}, B = {4, 5, 6, 7} and let f = {(1, 4), (2, 5), (3, 6)} be a function from A to B. Show that f is one-one.
In each of the following cases, state whether the function is one-one, onto or bijective. Justify your answer. (i) f : R → R defined by f (x) = 3 - 4x (ii) f : R → R defined by f (x) = 1 + x2
Let A and B be sets. Show that f : A × B → B × A such that f (a, b) = (b, a) is bijective function.
Let f : N → N be defined by f (n) = n n n n + 1/2 , , if is odd if is even for all n ∈ N. State whether the function f is bijective. Justify your answer.