Question

Let $f:N\to Y$ be a function defined as $f\left(x\right)=4x+3$, where, $Y=\{y\in N:y=4x+3forsomex\in N\}$, Show that f is invertible. Find the inverse.

Answer 1

$f\left(x\right)=4x+3$

$\to$ let $f\left(x_1\right)=f\left(x_2\right),x_1{x}_2\in N$

$4x_1+3=4x_2+3$

$\therefore 4x_1=4x_2$

$\therefore x_1=x_2$

Thus $f\left(x_1\right)=f\left(x_2\right)\Rightarrow x_1=x_2$ Hence the $fx$ is one-one.

Let $y\in y$ be a number of the form $y=4k+3$

$y=f\left(x\right)$

$\therefore 4k+3=4x+3$

$\therefore k=x$

$\to$ This corresponding to any $y\in y$ we have $x\in N$

The function is onto

the function being both one one and onto is invertible

$y=4x+3$

$\therefore y-3=4x$

$\therefore x=\frac{y-3}{4}$

$\therefore f\left(x\right)=\frac{x-3}{4}$ or $g\left(y\right)=\frac{4-3}{4}$ is inverse function.