Q. 64.0( 7 Votes )
Show that the function f : R → R defined by f(x) = 4x + 3 is invertible. Find the inverse of f.
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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