1.

Let `R^(+) ` be the set of all positive real numbers. Let `f : R^(+) to R^(+) : f(x) =e^(x) ` for all `x in R^(+)` . Show that f is invertible and hence find `f^(-1)`

Answer» f is one-one since
`f(x_(1)) =f(x_(2)) rArr e^(x) 1 rArr =e^(x)2 rArr x_(1)= x_(2)`
Now for each `y in R^(+)` there exists a positive real number namely log y such that
`f(log y) =e^(log y) =y`
`:. ` f is onto .
Thus , f is one-one onto and hence invertible .
we define :
`f^(-1) : R^(+) to R^(+) : f^(-1) (y) = " log y for all " y in R^(+)`


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