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

Answer 1

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