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

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