Let f : Z → Z : f(x) = 2x. Find g : Z → Z : g o f = IZ.

1.	Let f : Z → Z : f(x) = 2x. Find g : Z → Z : g o f = IZ.
Answer» To find: g : Z → Z : g o f = I_Z Formula used: (i) f o g = f(g(x)) (ii) g o f = g(f(x)) Given: (i) g : Z → Z : g o f = I_Z Solution: We have, f(x) = 2x Let f(x) = y ⇒ y = 2x \(\Rightarrow y=\frac{y}{2}\) \(\Rightarrow x=\frac{y}{2}\) let \(g(y)=\frac{y}{2}\) Where g: Z → Z For g o f, ⇒ g(f(x)) ⇒ g(2x) \(\Rightarrow \frac{2x}{2}\) ⇒ x = I_Z Clearly we can see that (g o f) = x = I_Z The required function is g(x) =\( \frac{2x}{2}\)

Answer»

To find: g : Z → Z : g o f = I_Z

Formula used: (i) f o g = f(g(x))

(ii) g o f = g(f(x))

Given: (i) g : Z → Z : g o f = I_Z

Solution: We have,

f(x) = 2x

Let f(x) = y

⇒ y = 2x

\(\Rightarrow y=\frac{y}{2}\)

\(\Rightarrow x=\frac{y}{2}\)

let \(g(y)=\frac{y}{2}\)

Where g: Z → Z

For g o f,

⇒ g(f(x))

⇒ g(2x)

\(\Rightarrow \frac{2x}{2}\)

⇒ x = I_Z

Clearly we can see that (g o f) = x = I_Z

The required function is g(x) =\( \frac{2x}{2}\)

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