1. Let f be a function defined by f(x) = \(\sqrt{x}\) is a function i

1.	1. Let f be a function defined by f(x) = \(\sqrt{x}\) is a function if it defined from (1)(a) f : N → N(b) f : R → R(c) f : R → R+(d) f : R+ → R+2. Check the injectivity and surjectivity of the following functions (3)(a) f : N → N defined by f(x) = x3(b) f : R → R given by f(x) = [x]
Answer» 1. (d) f : R⁺ → R⁺ 2. (a) For x, y ∈ N, f(x) = f(y) ⇒ x³ = y³ ⇒ x = y Therefore, f is injective For 2 ∈ N, there does not exist x in the domain N such that f(x) = x³ = 2. ∴ f is not surjective. (b) f : R → R given by f(x) = [x] It seen that f(1.1) = 1 and f(1.8) = 1; But 1.1 ≠ 1.8; ∴ f is not injective There does not exist any element x ∈ R such that f(x) = 0.7 ∴ f is not surjective.

1. Let f be a function defined by f(x) = \(\sqrt{x}\) is a function if it defined from (1)(a) f : N → N(b) f : R → R(c) f : R → R+(d) f : R+ → R+2. Check the injectivity and surjectivity of the following functions (3)(a) f : N → N defined by f(x) = x3(b) f : R → R given by f(x) = [x]

Answer»

1.
(d) f : R⁺ → R⁺

2.
(a) For x, y ∈ N,

f(x) = f(y) ⇒ x³ = y³ ⇒ x = y

Therefore, f is injective

For 2 ∈ N, there does not exist x in the domain N such that f(x) = x³ = 2.

∴ f is not surjective.

(b) f : R → R given by f(x) = [x]

It seen that f(1.1) = 1 and f(1.8) = 1;

But 1.1 ≠ 1.8;

∴ f is not injective

There does not exist any element x ∈ R

such that f(x) = 0.7

∴ f is not surjective.