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Let f: Q → Q: f(x) = 3x – 4. Show that f is invertible and find f -1.

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We know that

f(x1) = f(x2)

It can be written as

3x1 – 4 = 3x2 – 4

So we get

3x1 = 3x2

Where x1 = x2

f is one-one.

Take y = 3x – 4

It can be written as

y + 4 = 3x

So we get

x = (y + 4)/ 3

If y ∈ R there exists x = (y + 4)/ 3 ∈ R

We know that f (x) = f([y + 4]/ 3) = 3 ([y + 4]/ 3) – 4 = y

f is onto

Here, f is one-one onto and invertible.

Take y = f(x)

So we get

y = 4x – 3

It can be written as

x = (y + 4)/ 3

So f -1 (y) = (y + 4)/ 3

Hence, we define f -1: R → R: f -1(y) = (y + 4)/ 3 for all y ∈ R

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