Let $f:N \rightarrow Y$ be a function defined as $f(x)=4x+3$, where $Y=\{y \in N:y=4x+3 \text{ for some } x \in N\}$. Show that $f$ is invertible. Find its inverse.

$f$ is invertible; $f^{-1}(y)=\dfrac{y-3}{4}$

CBSE Class 12 · 2019 · Relations And Functions XIILong Answer

Let

f : N \to Y

be a function defined as

f (x) = 4 x + 3

, where

Y = {y \in N : y = 4 x + 3 for some x \in N}

. Show that

f

is invertible. Find its inverse.

Answer: $f$ is invertible; $f^{- 1} (y) = \frac{y - 3}{4}$

We show

f

is a bijection onto

Y

.
One-one:

f (x_{1}) = f (x_{2}) \Rightarrow 4 x_{1} + 3 = 4 x_{2} + 3 \Rightarrow x_{1} = x_{2}

.
Onto: by the definition of

Y

, every

y \in Y

has the form

y = 4 x + 3

for some

x \in N

, i.e.

y = f (x)

. So

f

is onto

Y

.
Hence

f

is invertible. To find the rule, set

y = 4 x + 3

and solve:

y = 4 x + 3 \Rightarrow x = \frac{y - 3}{4}

f^{- 1} (y) = \frac{y - 3}{4}, y \in Y

CBSE Class 12 Board · 2019 · Subjective