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Last updated at Jan. 28, 2020 by Teachoo
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Ex 1.2, 11 Let f: R → R be defined as f(x) = x4. Choose the correct answer. (A) f is one-one onto (B) f is many-one onto (C) f is one-one but not onto (D) f is neither one-one nor onto f(x) = x4 Checking one-one f (x1) = (x1)4 f (x2) = (x2)4 Putting f (x1) = f (x2) (x1)4 = (x2)4 x1 = x2 or x1 = –x2 Rough One-one Steps: 1. Calculate f(x1) 2. Calculate f(x2) 3. Putting f(x1) = f(x2) we have to prove x1 = x2 Since x1 does not have unique image, It is not one-one Eg: f(–1) = (–1)4 = 1 f(1) = (1)4 = 1 Here, f(–1) = f(1) , but –1 ≠ 1 Hence, it is not one-one Check onto f(x) = x4 Let f(x) = y , such that y ∈ R x4 = y x = ±𝑦^(1/4) Note that y is a real number, it can be negative also Putting y = −3 x = ±〖(−3)〗^(1/4) x = ± (√(−3))^(1/2) Which is not possible as root of negative number is not real Hence, x is not real ∴ f is not onto Hence, f is neither one-one nor onto Option D is correct
To prove one-one & onto (injective, surjective, bijective)
Onto function
One One and Onto functions (Bijective functions)
Example 7
Example 8
Example 9
Example 11 Important
Misc 5
Ex 1.2, 5 Important
Ex 1.2 , 6
Example 10
Ex 1.2, 1
Ex 1.2, 12
Ex 1.2 , 2 Important
Ex 1.2 , 7
Ex 1.2 , 11 You are here
Example 12 Important
Ex 1.2 , 9
Ex 1.2 , 3
Ex 1.2 , 4
Example 50
Example 51 Important
Ex 1.2 , 10 Important
Misc. 4 Important
Example 13 Important
Example 14 Important
Ex 1.2 , 8 Important
Example 46 Important
Misc 10 Important
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