Example 51
Last updated at May 29, 2018 by Teachoo
Transcript
Example 51 Consider a function f : [0, π2 ] → R given by f (x) = sin x and g: [0, π2 ] → R given by g(x) = cos x. Show that f and g are one-one, but f + g is not Checking one-one for f f : [0, π2 ] → R f (x) = sin x f(x1) = sin x1 f(x2) = sin x2 Putting f(x1) = f(x2) sin x1 = sin x2 So, x1 = x2 Hence, if f(x1) = f(x2) , then x1 = x2 ∴ f is one-one Checking one-one for g g : [0, π2 ] → R g(x) = cos x g(x1) = cos x1 g(x2) = cos x2 Putting g(x1) = g(x2) cos x1 = cos x2 So, x1 = x2 Hence, if g(x1) = g(x2) , then x1 = x2 ∴ g is one-one Checking one-one for f + g f + g : [0, π2 ] → R f + g (x) = sin x + cos x But (f + g) (0) = sin 0 + cos 0 = 0 + 1 = 1 &(f + g) 𝛑𝟐 = sin π2 + cos π2 = 1 + 0 = 1 Since, different elements 0, 𝜋2 have the same image 1, ∴ f + g is not one-one.
Examples
Example 1
Example 2
Example 3
Example 4 Important
Example 5
Example 6
Example 7
Example 8
Example 9
Example 10
Example 11 Important
Example 12 Important
Example 13 Important
Example 14
Example 15
Example 16
Example 17
Example 18 Important
Example 19 Important
Example 20
Example 21
Example 22
Example 23 Important
Example 24
Example 25 Important
Example 26
Example 27
Example 28 Important
Example 29
Example 30
Example 31
Example 32
Example 33
Example 34
Example 35
Example 36
Example 37
Example 38
Example 39
Example 40
Example 41
Example 42 Important
Example 43
Example 44
Example 45 Important
Example 46 Important
Example 47 Important
Example 48 Important
Example 49
Example 50
Example 51 You are here
Chapter 1 Class 12 Relation and Functions
Serial order wise
About the Author
Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 9 years. He provides courses for Maths and Science at Teachoo.