Find a bijection f such that f R rightarrow RSolutiona fxex

Find a bijection f such that f: R rightarrow R^+.

Solution

a)

f(x)=e^x is the required bijeciton

e^x is defined for all real numbers

And range of e^x is all positive real numbers

We can check it is bijective

Let, e^x=e^y

e^(x-y)=1

HEnce, x=y

HEce, e^x is one to one

Let, z be a positive real number

So, z=e^x means

x= log(z)

HEnce, f is onto