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

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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