Let f : A → A where A = {x : –1 ≤ x ≤ 1}. Find whether the following functions are bijective.
(i) x – sin x (ii) x |x|
(iii) tan 
(iv) x 4
Text Solution
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(i) No (ii) Yes (iii) Yes (iv) No
Sol. f : [–1, 1] → [–1, 1]
(i)
f(x) = x – sin x (odd function)
f ′ (x) = 1 – cos x ≥ 0 increasing function
Hence one - one
f(–1) = – 1 + sin 1
f(1) = 1 – sin 1
Range ≡ [–1 + sin 1, 1 – sin 1]
≠ co domain function is not bijecive
(ii)
f(x) = x|x| = 
one - one function
Range ≡ [–1, 1] = codomain
∴ onto function
(iii)
f(x) = tan 
by graph one-one onto
Bijective function
(iv) f(x) = x 4 even function
many-one ⇒ Not bijective
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