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Example 20 - Chapter 5 Class 12 Continuity and Differentiability
Last updated at April 16, 2024 by Teachoo
Continuity of composite functions
Chapter 5 Class 12 Continuity and Differentiability
Concept wise
- Checking continuity at a given point
- Checking continuity at any point
- Checking continuity using LHL and RHL
- Algebra of continous functions
-
Continuity of composite functions
- Checking if funciton is differentiable
- Finding derivative of a function by chain rule
- Finding derivative of Implicit functions
- Finding derivative of Inverse trigonometric functions
- Finding derivative of Exponential & logarithm functions
- Logarithmic Differentiation - Type 1
- Logarithmic Differentiation - Type 2
- Derivatives in parametric form
- Finding second order derivatives - Normal form
- Finding second order derivatives- Implicit form
- Proofs
- Verify Rolles theorem
- Verify Mean Value Theorem
Transcript
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 .
