Question

Abolition of intermediaries' and 'Land ceiling act' are a part ofLet f : N → Y be a function defined as f (x) = 4x + 3, where, Y = {y ∈ N: y = 4x + 3 for some x ∈ N}. Show that f is invertible. Find the inverse.

Give Explanation!​

Answer 1

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★Checking for Inverse :-

$\sf\mapsto{f(x) = 4x + 3}$

$\sf\mapsto{Let f(x) = y}$

$\sf\mapsto{y = 4x + 3}$

$\sf\mapsto{y – 3 = 4x}$

$\sf\mapsto{4x = y – 3}$

$\sf\mapsto{x = ( − 3)/4}$

Let g(y) = ( − 3)/4}

where g: Y → N

Now find gof :-

$\sf\mapsto{gof= g(f(x)}$

$\sf\mapsto{= g(4x + 3) }$

$\sf\mapsto{= [(4 + 3) − 3]/4}$

$\sf\mapsto{= [4 + 3 − 3]/4}$

$\sf\mapsto{=4x/4 = x = IN}$

Now find fog :-

$\sf\mapsto{fog= f(g(y))}$

$\sf\mapsto{= f [( − 3)/4] }$

$\sf\mapsto{=4[( − 3)/4] +3}$

$\sf\mapsto{= y – 3 + 3}$

$\sf\mapsto{= y + 0 = y = Iy}$

Thus, gof = INand fog = Iy,

Hence, f is invertible

Also, the Inverse of f = g(y) = [ – 3]/ 4

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Answer 2

Answer:

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Checking for Inverse :-

f(x) = 4x + 3

Let f(x) = y

y = 4x + 3

y – 3 = 4x

4x = y – 3

x = ( − 3)/4

Let g(y) = ( − 3)/4

where g: Y → N

Now find gof :-

gof= g(f(x))

= g(4x + 3) = [(4 + 3) − 3]/4

= [4 + 3 − 3]/4

=4x/4

= x = IN

Now find fog :-

fog= f(g(y))

= f [( − 3)/4]

=4[( − 3)/4] +3

= y – 3 + 3

= y + 0

= y = Iy

Thus, gof = INand fog = Iy,

Hence, f is invertible

Also, the Inverse of f = g(y) = [ – 3]/ 4

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