Q. 64.0( 7 Votes )

Show that the fun

Answer :

We have f : R R and f(x) = 4x + 3.


Recall that a function is invertible only when it is both one-one and onto.


First, we will prove that f is one-one.


Let x1, x2ϵ R (domain) such that f(x1) = f(x2)


4x1 + 3 = 4x2 + 3


4x1 = 4x2


x1 = x2


So, we have f(x1) = f(x2) x1 = x2.


Thus, function f is one-one.


Now, we will prove that f is onto.


Let y ϵ R (co-domain) such that f(x) = y


4x + 3 = y


4x = y – 3



Clearly, for every y ϵ R, there exists x ϵ R (domain) such that f(x) = y and hence, function f is onto.


Thus, the function f has an inverse.


We have f(x) = y x = f-1(y)


But, we found f(x) = y


Hence,


Thus, f(x) is invertible and


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