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In each of the following cases, state whether the function is one-one, onto or bijective. Justify your answer. (i) f : R → R defined by f(x) = 3 – 4x (ii) f : R → R defined by f(x) = 1 + x^2

Class 12, NCERT Chapter 1,  Exercise 1.2, Q7

(i) 
Here f(x) = 3 – 4x
Then f(x1) = 3 – 4x1
and f (x2) = 3 – 4x2
Now f(x1) = f (x2)
⇒ 3 – 4x1 = 3 – 4x2
⇒ – 4x1 = –4x2
⇒ x1 = x2
∴ f(x1) = f(x2)
& x1 = x2
So, f is one-one function.
Let f(x) = y ∈ R
Then y=3 - 4x
⇒ x= 3-y/4
(ii)
Let x1 = 2 and x2 = –2 ∈ R
Here f(x) = 1 + x2
Then f(x1) = f (2) = 1 + (2) 2 = 5
and f (x2) = f(–2) = 1 + (–2) = 5
∴ f (x1) = f(x2) but x1 ≠ x2
So, f is not one-one function.
Let f(x) = – 2 ⇒R
Then 1 + x2 = –2
⇒ x2 = –3 = ± √–3 ∈R
So, f is not onto function.

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