1.

Find the domain and range of the function `f(x) = sin^(-1)((1+e^(x))^(-1))`.

Answer» `f(x) = sin^(-1)((1+e^(x))^(-1))`
We know that `e^(x) gt 0` for all real x.
` :. e^(x) +1 gt 1`
`implies 0 lt (1)/(1+e^(x)) lt 1`
So, f(x) is defined for all real values of x.
Hence domain is R.
Also `0 lt (1)/(1+e^(x)) lt 1`
`implies sin^(-1)0 lt "sin"^(-1)(1)/(1+e^(x)) lt sin^(-1)1`
`implies 0 lt "sin"^(-1)(1)/(1+e^(x)) lt (pi)/(2)`
Therefore, range of the function is `(0, (pi)/(2))`


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