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Let f:RR be defined as f(x)=10x+7. Find the function g:RR such that g o f=f g=1R.
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Solution:

It is given that f:RR is defined as f(x)=10x+7.

One-one:

Let f(x)=f(y), where x,yR.

10x+7=10y+7

x=y

f is a one-one function.

Onto:

For yR

let y=10x+7.

x=y710R Therefore, for any yR, there exists x=y710R

such that, f(x)=f(y710)=10(y710)+7=y7+7=y.

f is onto.

Therefore, f is one-one and onto.

Thus, f is an invertible function.

Let us define,

g:RR

as

g(y)=y710.

Now, we have:

gf(x)=g(f(x))=g(10x+7)=(10x+7)710=10x10=10

And,

fg(y)=f(g(y))=f(y710)=10(y710)+7=y7+7=ygf=IR and fg=IR

Hence, the required function g:RR is defined as g(y)=y710

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