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

$Given, f : R \to R Since g (x) = 2 x is a polynomial, g : R \to R Clearly, g o f : R \to R a n d f + f : R \to R So, domains of gof and f+f are the same. (g o f) (x) = g (f (x)) = 2 f (x) (f + f) (x) = f (x) + f (x) = 2 f (x) ⇒ (g o f) (x) = (f + f) (x), \forall x \in R Hence, g o f = f + f$

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Let $f (x) = x^{2}$ and $g (x) = 2 x + 1$ be two real functions. Find (f + g), (f - g) (x), (fg) (x) and $(\frac{f}{g}) (x)$

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