Question

A function f : R → R is defined by f(x) = x². Determine (a) range of f, (b) {x : f(x) = 4}, (c) [y : f(y) = −1].

Answer 1

(a) Given:
f (x) = x²
Range of f = R⁺ (Set of all real numbers greater than or equal to zero)

(b) Given:
f (x) = x²
⇒ x² = 4
⇒ x = ± 2
∴ {x : f (x) = 4 } = { $-$ 2, 2}.

(c) { y : f (y) = $-$1}
⇒ f (y) = $-$1
It is clear that x² = $-$1 but x² ≥ 0 .
⇒ f (y) ≠ $-$1
∴ {y : f (y) = $-$1} = Φ