Q. 123.8( 5 Votes )

Show that the exponential function f: R R, given by f(x) = ex, is one – one but not onto. What happens if the co – domain is replaced by R0 + (set of all positive real numbers).

Answer :

TIP: One – One Function: – A function is said to be a one – one functions or an injection if different elements of A have different images in B.


So, is One – One function


a≠b


f(a)≠f(b) for all


f(a) = f(b)


a = b for all


Onto Function: – A function is said to be a onto function or surjection if every element of A i.e, if f(A) = B or range of f is the co – domain of f.


So, is Surjection iff for each, there exists such that f(a) = b


Now, given by f(x) = ex


Check for Injectivity:


Let x,y be elements belongs to R i.e such that


So, from definition


f(x) = f(y)


ex = ey



ex – y = 1


ex – y = e0


x – y = 0


x = y


Hence f is One – One function


Check for Surjectivity:


Here range of f = (0,∞) ≠ R


Therefore f is not onto


Now if co – domain is replaced by R0 + (set of all positive real numbers) i.e (0,∞) then f becomes an onto function.


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