Question:
Let $f: Z \rightarrow Z: f(x)=2 x$. Find $g: Z \rightarrow Z: g \circ f=l_{Z}$.
Solution:
To find: $g: Z \rightarrow Z: g \circ f=I_{Z}$
Formula used: (i) $f \circ g=f(g(x))$
(ii) g o f = g(f(x))
Given: (i) $g: Z \rightarrow Z:$ g of $=I_{Z}$
Solution: We have,
$f(x)=2 x$
Let $f(x)=y$
⇒ y = 2x
$\Rightarrow \mathrm{x}=\frac{\mathrm{y}}{2}$
$\Rightarrow \mathrm{x}=\frac{\mathrm{y}}{2}$
Let $g(y)=\frac{y}{2}$
Where g: Z → Z
For g o f,
⇒ g(f(x))
⇒ g(2x)
$\Rightarrow \frac{2 x}{2}$
$\Rightarrow x=I_{Z}$
Clearly we can see that $(g \circ f)=x=l z$
The required function is $g(x)=\frac{x}{2}$