Example 20 - Chapter 5 Class 12 Continuity and Differentiability
Last updated at Dec. 16, 2024 by Teachoo
Last updated at Dec. 16, 2024 by Teachoo
Example 20 Show that the function f defined by f (x) = |1− 𝑥 + | 𝑥 ||, where x is any real number is a continuousGiven 𝑓(𝑥) = |(1−𝑥+|𝑥|)| Let 𝒈(𝒙) = 1−𝑥+|𝑥| & 𝒉(𝒙) = |𝑥| Then , 𝒉𝒐𝒈(𝒙) = ℎ(𝑔(𝑥)) = ℎ(1−𝑥+|𝑥|) = |(1−𝑥+|𝑥|)| = 𝒇(𝒙) We know that, Modulus function is continuous ∴ 𝒉(𝒙) = |𝑥| is continuous Also, 𝒈(𝒙) = (𝟏−𝒙)+|𝒙| Since (1−𝑥) is a polynomial & every polynomial function is continuous ∴ (𝟏−𝒙) is continuous Also, |𝒙| is also continuous Since Sum of two continuous function is also continuous Thus, 𝑔(𝑥) = 1−𝑥+|𝑥| is continuous . Hence, 𝑔(𝑥) & ℎ(𝑥) are both continuous . We know that If two function of 𝑔(𝑥) & ℎ(𝑥) both continuous, then their composition 𝒉𝒐𝒈(𝒙) is also continuous Hence, 𝒇(𝒙) is continuous .
Examples
Example 2
Example 3
Example 4 Important
Example 5
Example 6
Example 7
Example 8
Example 9
Example 10
Example 11 Important
Example 12
Example 13 Important
Example 14
Example 15 Important
Example 16
Example 17 Important
Example 18
Example 19
Example 20 Important You are here
Example 21
Example 22
Example 23
Example 24 Important
Example 25
Example 26 (i)
Example 26 (ii) Important
Example 26 (iii) Important
Example 26 (iv)
Example 27 Important
Example 28
Example 29 Important
Example 30 Important
Example 31
Example 32
Example 33 Important
Example 34 Important
Example 35
Example 36 Important
Example 37
Example 38 Important
Example 39 (i)
Example 39 (ii) Important
Example 40 (i)
Example 40 (ii) Important
Example 40 (iii) Important
Example 41
Example 42 Important
Example 43
Question 1
Question 2 Important
Question 3
Question 4 Important
Question 5
Question 6 Important
About the Author
Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science and Computer Science at Teachoo