Indicate whether each of the following function is invertible in the giveninterval. Explain.  cos(ln x) on (0, e^pi ]    

Expert Answers
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You need to test if the function `cos(ln x)` is invertible in `(0, e^pi]` , hence, you need to perform the following steps, such that:

 

`y = cos(ln x) => cos^(-1) y = cos^(-1) cos(ln x) cos^(-1) y = ln x => x = e^(cos^(-1) y) `

Since `x > 0` , hence, you may consider `x = 1` , such that:

`1 = e^(cos^(-1) y) => e^0 = e^(cos^(-1) y) => 0 = cos^(-1) y => y = 1`

Considering` x = e^p` i yields:

`e^pi = e^(cos^(-1) y) => pi = cos^(-1) y => y = -1`

Hence, testing if the function `y = cos(ln x)` is invertible in `(0, e^pi]` , yields that it is, `f^(-1)(x) = e^(cos^(-1) x).`