Find whether the following functions are one-one or many-one & into or onto if f : D → R where D is its domain.
(i) f(x) = |x 2 + 5x + 6 | (ii) f(x) = | nx|
(iii) f(x) = 
(iv) f(x) = x + 
, x ∈ (0, ∞ )
(v)f(x) = 
(vi) f(x) = 
– cos π x
(vii) f(x) = 
(viii) f(x) = x cos x(ix) f(x) = 
Text Solution
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(i) many-one & into (ii) many-one & into
(iii) one-one & onto (iv) many-one & into
(v) one – one & into (vi) many-one & into
(vii) many-one & into (viii) many-one & onto
(ix) many-one & into
Sol. (i) y = |(x + 2) (x + 3)|
many - one function
(ii) y = | nx|
many - one function
(iii) f(x) =sin 4x, x ∈ 
period = 
one-one function
(iv) f(x) = x + 
, x 
∈ (0, ∞ )
many one function
(v) f(x) = 
⇒ f ′ = 
increasing function
Hence one - one
(vi) f(x) = 
even function
Hence many - one
(vii) f(x) = x 3 + 
⇒ f ′ (x) = 3x 2 – 
= 0 ⇒ x = ±1
Also f(x) ≠ 0 ∴ Range 
R ∴ f(x) is into function.
(viii) f(x) = x cos x odd function (f(0) = 0) ⇒ f ′ (x) = cos x – x sin x
f(x) is an odd continuous function for which 
= ∞ . Hence range ∈ R onto function
(ix) f(x) = 
Clearly many one. Clearly f(x) ∈ (– ∞ , –1] ∪ [1, ∞ ) ⇒ Range 
R into function
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