# Let f : N <span l

We have f : N N and f(x) = 9x2 + 6x – 5.

We need to prove f : N S is invertible.

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 N (domain) such that f(x1) = f(x2)

9x12 + 6x1 – 5 = 9x22 + 6x2 – 5

9x12 + 6x1 = 9x22 + 6x2

9x12 – 9x22 + 6x1 – 6x2 = 0

9(x12 – x22) + 6(x1 – x2) = 0

9(x1 – x2)(x1 + x2) + 6(x1 – x2) = 0

(x1 – x2)[9(x1 + x2) + 6] = 0

x1 – x2 = 0 (as x1, x2ϵ R+)

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 S (co-domain) such that f(x) = y

9x2 + 6x – 5 = y

Adding 6 to both sides, we get

9x2 + 6x – 5 + 6 = y + 6

9x2 + 6x + 1 = y + 6

(3x + 1)2 = y + 6

Clearly, for every y ϵ S, there exists x ϵ N (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

Hence, we have

Thus, f-1(43) = 2 and f-1(163) = 4.

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