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# Check the injectivity and surjectivity of the following functions: (i). f : N → N given by f (x) = x^{2 }(ii). f : Z → Z given by f (x) = x^{2 }(iii). f : R → R given by f (x) = x^{2 }(iv). f : N → N given by f (x) = x^{3 }(v). f : Z → Z given by f (x) = x^{3}

**Solution:**

The function is said to be injective, or one to one, if each element of the codomain of the given function is mapped with at most one element of the domain.

∈ represents belongs to.

i.

For f : N → N given by f (x) = x^{2}

x, y ∈ N f (x) = f (y) ⇒ x^{2} = y^{2} ⇒ x = y

Therefore,

f is injective.

2 ∈ N. But, there does not exist any x in N such that f (x) = x^{2} = 2

⇒ f is not surjective

Function f is injective but not surjective.

ii.

f : Z → Z given by f (x) = x^{2}

f (- 1) = f (1) = 1 but - 1 ≠ 1

⇒ f is not injective.

- 2 ∈ Z. But, there does not exist any x ∈ Z such that f (x) = - 2 ⇒ x^{2} = - 2

⇒ f is not surjective.

Function f is neither injective nor surjective.

iii.

f : R → R given by f (x) = x^{2}

f (- 1) = f (1) = 1 but - 1 ≠ 1

⇒ f is not injective.

- 2 ∈ Z. But, there does not exist any x ∈ Z such that f (x) = - 2 ⇒ x^{2} = - 2

⇒ f is not surjective.

Function f is neither injective nor surjective.

iv.

f : N → N given by f (x) = x^{3}

x, y ∈ N

f (x) = f (y) ⇒ x^{3} = y^{3} ⇒ x = y

⇒ f is injective.

2 ∈ N. But, there does not exist any x in N such that f (x) = x^{3} = 2

⇒ f is not surjective

Function f is injective but not surjective.

v.

f : Z → Z given by f (x) = x^{3}

x, y ∈ Z

f (x) = f (y) ⇒ x^{3} = y^{3} ⇒ x = y

Therefore,

f is injective.

2 ∈ Z . But, there does not exist any x in Z such that f (x) = x^{3} = 2

Therefore,

f is not surjective.

Function f is injective but not surjective

NCERT Solutions for Class 12 Maths - Chapter 1 Exercise 1.2 Question 2

## Check the injectivity and surjectivity of the following functions: (i) f : N → N given by f(x) = x^{2} (ii) f : Z → Z given by f(x) = x^{2} (iii) f : R → R given by f(x) = x^{2} (iv) f : N → N given by f(x) = x^{3} (v) f : Z → Z given by f(x) = x^{3}

**Summary:**

(i) Function f is injective but not surjective. (ii) Function f is neither injective nor surjective. (iii) Function f is neither injective nor surjective. (iv) Function f is injective but not surjective. (v) Function f is injective but not surjective

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