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Question
Let f be any real function and let g be a function given by g(x) = 2x. Prove that gof = f + f.
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Solution
Given, f : R → R
Since g(x) = 2x is a polynomial, g : R → R
Clearly, gof : R → R and f + f : R → R
So, domains of gof and f+f are the same.
(gof) (x) = g (f (x)) = 2 f (x)
(f+f) (x) = f (x) + f (x) = 2 f (x)
⇒ ( gof ) (x) = (f+f) (x), ∀x ∈ R
Hence, gof = f + f
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