The function `f(x)=e^(x),x""inR` is

Options

A. onto but not one-one
B. one-one onto
C. one-one but onto
D. neither one-one nor onto

Answer

C

Explanation

Let `f(x)=f(y)`
To show that f(x) is one-one We have to show that x=y
Now, f(x)=f(y)
`impliese^(x)=e^(y)implies(e^(x))/(e^(y))=1impliese^(x-y)=1`
Take log on both side
`loge^(x-y)=log1`
`impliesx-y=0impliesx=y`
Hence f(x) one-one `AAx""inR`

-2 is an element of the co-domain R. There doesnot exist any element X in the domain R such that `-2=e^(x)=f(x)`.
Hence, by definition, f is not a onto function.