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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)? - Mathematics

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Question

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)?

Sum
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Solution

f : R → R, given by f(x) = ex

Injectivity:
Let x and y be any two elements in the domain (R), such that f(x) = f(y)
 f(x)=f(y)

⇒ `e^x = e^y`

⇒  x = y

So, f is one-one .

Surjectivity:
We know that range of ex is (0, ∞) = R+
Co-domain = R
Both are not same.
So, f is not onto.

If the co-domain is replaced by R+, then the co-domain and range become the same and in that case, f is onto and hence, it is a bijection.

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Chapter 2: Functions - Exercise 2.1 [Page 32]

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RD Sharma Mathematics [English] Class 12
Chapter 2 Functions
Exercise 2.1 | Q 12 | Page 32

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