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Show that the function f: R → R: f(x) = 2x + 3 is invertible and find f -1.

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We know that

f(x1) = f(x2)

It can be written as

2x1 + 3 = 2x2 + 3

On further calculation

2x1 = 2x2

So we get

x1 = x2

Hence, f is one-one.

Consider y = 2x + 3

It can be written as

y – 3 = 2x

So we get

x = (y – 3)/ 2

If y ∈ R, there exists x = (y – 3)/ 2 ∈ R

f (x) = f ([y-3]/2) = 2([y – 3]/ 2) +3 = y

f is onto

Here, f is one-one onto and invertible.

Take y = f(x)

It can be written as

y = 2x + 3

So we get

x = (y-3)/ 2

So f -1 (y) = (y – 3)/ 2

Hence, we define f -1: R → R: f -1(y) = (y – 3)/ 2 for all y ∈ R

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