if `f:RtoR` is defined as `f(x)=e^(|x|)-e^(-x)` then the correct statements(s) is/are

Options

A. `f` is one- one onto function
B. `f` is many one into function
C. range of `f` is `[0,infty]`
D. range of `f` is `(-infty,infty)`

Answer

B, C

Explanation

`f(x)=e^(|x|)-e^(-x)=[{:(e^(x)-e^(-x),xge0),(0,xlt0):}`
for `xge0` `f(x)=e^(x)-e^(-x)`
`impliesf'(x)=e^(x)+e^(-x)gt0,Aaxxge0`
`impliesf(x)` is st. increasing in `[0,infty]`
`because` range of `f(x)` in `[0,infty]` is `[f(0),f(infty))`
i.e., range is `[0,infty]`
for `xle0` `f(x)=0` ltbr `implies` many one function and range of `f(x)` is `[0,infty]`