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Example 51 - f(x) = sin x, g(x) = cos x. Show f, g are one-one - Examples

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  1. Chapter 1 Class 12 Relation and Functions
  2. Serial order wise
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Example 51 Consider a function f : [0, π﷮2﷯ ] → R given by f (x) = sin x and g: [0, π﷮2﷯ ] → 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, π﷮2﷯ ] → 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, π﷮2﷯ ] → 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, π﷮2﷯ ] → R f + g (x) = sin x + cos x But (f + g) (0) = sin 0 + cos 0 = 0 + 1 = 1 &(f + g) 𝛑﷮𝟐﷯﷯ = sin π﷮2﷯ + cos π﷮2﷯ = 1 + 0 = 1 Since, different elements 0, 𝜋﷮2﷯ have the same image 1, ∴ f + g is not one-one.

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He provides courses for Mathematics from Class 9 to 12. You can ask questions here.
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