If f : R → R, f(x) = x2, then find(i) Range of f,(ii) {x

1.	If f : R → R, f(x) = x2, then find(i) Range of f,(ii) {x \| f(x) = 4},(iii) {y \| f(y) = -1}
Answer» (i) Given, f : R → R and f(x) = x² then if x < 0 ⇒ x² > 0 x = 0 ⇒ x² = 0 x > 0 ⇒ x² > 0 So, f(x) = x² ≥ 0 ∀ x ∈ R Hence, range of R = R⁺ ∪ {0} or {x ∈ R \| 0 ≤ x < ∞} (ii) f(x) = 4 ⇒ x² = 4 ⇒ x = ±2 Hence, {x : y (x) = 4} = {-2, 2}. (iii) f(y) = -1 ⇒ y² = -1 ⇒ y = ±√1 ⇒ {y : f(y) = -1} = Φ null set.

If f : R → R, f(x) = x2, then find(i) Range of f,(ii) {x | f(x) = 4},(iii) {y | f(y) = -1}

Answer»

(i) Given, f : R → R

and f(x) = x²

then if x < 0 ⇒ x² > 0

x = 0 ⇒ x² = 0

x > 0 ⇒ x² > 0

So, f(x) = x² ≥ 0 ∀ x ∈ R

Hence, range of R = R⁺ ∪ {0} or {x ∈ R | 0 ≤ x < ∞}

(ii) f(x) = 4

⇒ x² = 4

⇒ x = ±2

Hence, {x : y (x) = 4} = {-2, 2}.

(iii) f(y) = -1

⇒ y² = -1

⇒ y = ±√1

⇒ {y : f(y) = -1} = Φ null set.