# 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).

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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