One One / Many One

f(x) = , Domain x ∈ R ⇒ f ′ (x) =

f ′ (x) = > 0 ⇒ x ∈ (– ∞ , 0) ∪

f ′ (x) < 0 ⇒ x ∈ ⇒ f ′ (x) = 0 ⇒ x = 0,

Function is increasing and decreasing in different intervals, so non monotonic

∴ Many one function.

Onto / Into

f(x) = ⇒ 2x 2 – x + 5 > 0, ∀ x ∈ R and 7x 2 + 2x + 10 > 0 ∀ x ∈ R

 a = 2 > 0 and  a = 7 and D = 4 – 280 < 0

D = 1 – 40 = – 39 < 0 ∴ f(x) > 0 ∀ x ∈ R

Also f(x) never tends to ± ∞ as 7x 2 + 2x + 10 has no real roots, Range ≠ Codomain so into function.

──────────────────────────────────────────────────────────────────────────────────────────