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Let f:R→R be defined as f(x)=10x+7. Find the function g:R→R such that g o f=f ∘g=1R.
1 Answer
written 2.9 years ago by | • modified 2.9 years ago |
Solution:
It is given that f:R→R is defined as f(x)=10x+7.
One-one:
Let f(x)=f(y), where x,y∈R.
⇒10x+7=10y+7
⇒x=y
∴f is a one-one function.
Onto:
For y∈R
let y=10x+7.
⇒x=y−710∈R Therefore, for any y∈R, there exists x=y−710∈R
such that, f(x)=f(y−710)=10(y−710)+7=y−7+7=y.
∴f is onto.
Therefore, f is one-one and onto.
Thus, f is an invertible function.
Let us define,
g:R→R
as
g(y)=y−710.
Now, we have:
g∘f(x)=g(f(x))=g(10x+7)=(10x+7)−710=10x10=10
And,
f∘g(y)=f(g(y))=f(y−710)=10(y−710)+7=y−7+7=y∴g∘f=IR and f∘g=IR
Hence, the required function g:R→R is defined as g(y)=y−710