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Question
Let f be a function from R to R, such that f(x) = cos (x + 2). Is f invertible? Justify your answer.
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Solution
Injectivity:
Let x and y be two elements in the domain (R), such that
f (x) = f (y)
⇒ cos ( x+2 ) = cos ( y+2 )
⇒ x+2 = y+2 or x + 2 = 2π − ( y+2 )
⇒ x = y or x + 2 = 2π - y - 2
⇒ x = y or x = 2π - y - 4
So, we cannot say that x = y
For example,
`cos π/2 = cos (3π)/2 =0 `
`So, π /2and (3x)/2` have the same image 0.
f is not one-one.
f is not a bijection.
Thus, f is not invertible.
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