Find the value

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 $g(x)=\frac{x}{2}$

 

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