Arts (English Medium) इयत्ता १२ - CBSE Question Bank Solutions for Mathematics

Check the injectivity and surjectivity of the following function:

f : N → N given by f(x) = x³

Check the injectivity and surjectivity of the following function:

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) = {(1", if"  x > 0), (0", if"  x  = 0), (-1", if"  x < 0):}` 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 the following case, state whether the function is one-one, onto or bijective. Justify your answer.

f : R → R defined by f(x) = 3 − 4x

In the following case, state whether the function is one-one, onto or bijective. Justify your answer.

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 a bijective function.

Let f : N → N be defined by f(n) = `{((n+1)/2", if n is odd"),(n/2", if n is even"):}` for all n ∈ N.

State whether the function f is bijective. Justify your answer.

Let A = R − {3} and B = R − {1}. Consider the function f : A → B defined by f(x) = `((x- 2)/(x -3))`. Is f one-one and onto? Justify your answer.

Let f : R → R be defined as f(x) = x⁴. Choose the correct answer.

Let f : R → R be defined as f(x) = 3x. Choose the correct answer.

Let f: R → R be defined as f(x) = 10x + 7. Find the function g: R → R such that g o f = f o g = 1_R.

Show that the function f : R → {x ∈ R : −1 < x < 1} defined by f(x) = `x/(1 + |x|)`, x ∈ R is one-one and onto function.

Show that the function f : R → R given by f(x) = x³ is injective.

Give examples of two functions f: N → Z and g: Z → Z such that g o f is injective but gis not injective.

(Hint: Consider f(x) = x and g(x) =|x|)

Given examples of two functions f: N → N and g: N → N such that gof is onto but f is not onto.

(Hint: Consider f(x) = x + 1 and `g(x) = {(x-1, ifx >1),(1, if x = 1):}`

Find the number of all onto functions from the set {1, 2, 3, ..., n} to itself.

Let S = {a, b, c} and T = {1, 2, 3}. Find F⁻¹ of the following functions F from S to T, if it exists.

F = {(a, 3), (b, 2), (c, 1)}