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Question
If f, g, h are real valued functions defined on R, then prove that (f + g) o h = f o h + g o h. What can you say about f o (g + h)? Justify your answer
Solution
Given f: R → R
g: R → R and
h: R → R (f + g)
oh: R → R and
(f o h + g o h): R → R for any x ∈ R.
[(f + g)oh(x) = (f + g)h(x)
= f(h(x)) + g(h(x))
= foh(x) + goh(x)
∴ (f + g)oh = foh + goh
Also fo(g + h)(x) = f((g + h)(x)) for any x ∈ R
= f(g(x) + h(x))
= f(g(x)) + f(h(x))
= fog(x) + foh(x)
∴ fo(g + h) = fog + foh
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